PEACOCK 


BOOLE 


SYLVESTER 


DE  MORGAN 


CAYLEY 


SMIT 


HAMILTON 


CLIFFORD 


KlRKMAN 


TODHUNTER 


MATHEMATICAL    MONOGRAPHS 

EDITED    BY 

MANSFIELD  MERRIMAN  AND  ROBERT  S.  WOODWARD 


No.  17 .      •••"•      " 

LECTURES    ON 

TEN  BRITISH   MATHEMATICIANS 

OF  THE  NINETEENTH  CENTURY 


ALEXANDER  MACFARLANE, 

LATE  PRESIDENT  OF  THE  INTERNATIONAL  ASSOCIATION  FOR  PROMOTING 
THE  STUDY  OF  QUATERNIONS 


FIRST    EDITION 
FIRST   THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:    CHAPMAN   &   HALL,  LIMITED 

1916 


MATHEMATICAL  MONOGRAPHS 

EDITED    BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 

Octavo,  Cloth. 

No.     1.     History    of     Modern     Mathematics.      By 

DAVID  EUGENE  SMITH.     $1.00  net. 

No.     2.     Synthetic       Projective     Geometry.         By 

GEORGE  BRUCE  HALSTED.     |i.oo  net. 

No.     3.     Determinants.     By  LAENAS  GIFFORD  WELD. 
$1.00  net. 

No.     4.     Hyperbolic   Functions.        By   JAMES    Mc- 
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No.     6.     Grassmann's  Space  Analysis.    By  EDWARD 
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No.     7.     Probability    and    Theory    of    Errors.     By 

ROBERT  S.  WOODWARD.     Ji.oo  net. 

No.     8.     Vector     Analysis    and    Quaternions.     By 

ALEXANDER  MACFARLANE.     Ji.oo  net. 

No.     9.     Differential     Equations.         BY      WILLIAM 
WOOLSEY  JOHNSON.     Ji.oo  net. 

No.  10.     The  Solution  of  Equations.  By  MANSFIELD 
MERRIMAN.     Ji.oo  net. 

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CARMICHAEL.     Ji.oo  net. 

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DICKSON.     $1.25  net. 

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HENDERSON.     ,81.25  net. 

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No.  17.     Ten  British    Mathematicians.     By  ALEX- 
ANDER MACFARLANE.     $1.25  net. 

PUBLISHED  BY 

JOHN  WILEY  &  SONS,  Inc.,  NEW  YORK. 
CHAPMAN  &  HALL,  Limited,  LONDON 


No.  17  COPYRIGHTED,  1916, 

BY 

HELEN  S.  MACFARLANE 


IS 
M3 


PREFACE 


DURING  the  years  1901-1904  Dr.  Alexander  Macfarlane 
delivered,  at  Lehigh  University,  lectures  on  twenty-five  British 
mathematicians  of  the  nineteenth  century.  The  manuscripts 
of  twenty  of  these  lectures  have  been  found  to  be  almost  ready 
for  the  printer,  although  some  marginal  notes  by  the  author 
indicate  that  he  had  certain  additions  in  view.  The  editors 
have  felt  free  to  disregard  such  notes,  and  they  here  present 
ten  lectures  on  ten  pure  mathematicians  in  essentially  the  same 
form  as  delivered.  In  a  future  volume  it  is  hoped  to  issue 
lectures  on  ten  mathematicians  whose  main  work  was  in  physics 
and  astronomy. 

These  lectures  were  given  to  audiences  composed  of  students, 
instructors  and  townspeople,  and  each  occupied  less  than  an 
hour  in  delivery.  It  should  hence  not  be  expected  that  a  lecture 
can  fully  treat  of  all  the  activities  of  a  mathematician,  much 
less  give  critical  analyses  of  his  work  and  careful  estimates  of 
his  influence.  It  is  felt  by  the  editors,  however,  that  the  lectures 
will  prove  interesting  and  inspiring  to  a  wide  circle  of  readers 
who  have  no  acquaintance  at  first  hand  with  the  works  of  the 
men  who  are  discussed,  while  they  cannot  fail  to  be  of  special 
interest  to  older  readers  who  have  such  acquaintance. 

It  should  be  borne  in  mind  that  expressions  such  as  "  now," 
"  recently,"  "  ten  years  ago,"  etc.,  belong  to  the  year  when  a 
lecture  was  delivered.  On  the  first  page  of  each  lecture  will 
be  found  the  date  of  its  delivery. 

For  six  of  the  portraits  given  in  the  frontispiece  the 
editors  are  indebted  to  the  kindness  of  Dr.  David  Eugene 
Smith,  of  Teachers  College,  Columbia  University. 

Z 


4  PREFACE 

Alexander  Macfarlane  was  born  April  21, 1851,  at  Blairgowrie, 
Scotland.  From  1871  to  1884  he  was  a  student,  instructor  and 
examiner  in  physics  at  the  University  of  Edinburgh,  from  1885 
to  1894  professor  of  physics  in  the  University  of  Texas,  and  from 
1895  to  1908  lecturer  in  electrical  engineering  and  mathematical 
physics  in  Lehigh  University.  He  was  the  author  of  papers  on 
algebra  of  logic,  vector  analysis  and  quaternions,  and  of  Mono- 
graph No.  8  of  this  series.  He  was  twice  secretary  of  the  sec- 
tion of  physics  of  the  American  Association  for  the  Advancement 
of  Science,  and  twice  vice-president  of  the  section  of  mathematics 
and  astronomy.  He  was  one  of  the  founders  of  the  International 
Association  for  Promoting  the  Study  of  Quaternions,  and  its 
president  at  the  time  of  his  death,  which  occured  at  Chatham, 
Ontario,  August  28,  1913.  His  personal  acquaintance  with 
British  mathematicians  of  the  nineteenth  century  imparts  to 
many  of  these  lectures  a  personal  touch  which  greatly  adds 
to  their  general  interest. 


ALEXANDER  MACFARLANE 
From   a    photograph    of    189 


CONTENTS 


PORTRAITS  OF  MATHEMATICIANS Frontispiece 

GEORGE  PEACOCK  (1791-1858) PAGE      7 

A  Lecture  del. vered  Apri  12,1901. 

AUGUSTUS  DE  MORGAN  (1806-1871) 19 

A  Lecture  delivered  April  13,  1901. 

SIR  WILLIAM  ROWAN  HAMILTON  (1805-1865) 34 

A  Lecture  delivered  April  16,  1901. 

GEORGE  BOOLE  (1815-1864) 50 

A  Lecture  de  ivered  April  19,  1901. 

ARTHUR  CAYLEY  (1821-1895) 64 

A  Lecture  delivered  April  20/1901. 

WILLIAM  KINGDON  CLIFFORD  (1845-1879) 78 

A  Lecture  delivered  April  23,  1901. 

HENRY  JOHN  STEPHEN  SMITH  (1826-1883) 92 

A  Lecture  delivered  March  15,  1902. 

JAMES  JOSEPH  SYLVESTER  (1814-1897) 107 

A  Lecture  delivered  March  21,  1902. 

THOMAS  PENYNGTON  KIRKMAN  (1806-1895 I22 

A  Lecture  delivered  April  20,  1903. 

ISAAC  TODHUNTER  (1820-1884) I3A 

A  Lecture  de  ivered  April  13,  1904. 

INDEX 147 

5 


TEN   BRITISH  MATHEMATICIANS 


GEORGE  PEACOCK  * 

(1791-1858) 

GEORGE  PEACOCK  was  born  on  April  9,  1791,  at  Den  ton 
in  the  north  of  England,  14  miles  from  Richmond  in  Yorkshire. 
His  father,  the  Rev.  Thomas  Peacock,  was  a  clergyman  of  the 
Church  of  England,  incumbent  and  for  50  years  curate  of  the 
parish  of  Denton,  where  he  also  kept  a  school.  In  early  life 
Peacock  did  not  show  any  precccity  of  genius,  and  was  more 
remarkable  for  daring  feats  of  climbing  than  for  any  special 
attachment  to  study.  He  received  his  elementary  education 
from  his  father,  and  at  17  years  of  age,  was  sent  to  Richmond, 
to  a  school  taught  by  a  graduate  of  Cambridge  University  to 
receive  instruction  preparatory  to  entering  that  University. 
At  this  school  he  distinguished  himself  greatly  both  in  classics 
and  in  the  rather  elementary  mathematics  then  required  for 
entrance  at  Cambridge.  In  1809  he  became  a  student  of  Trinity 
College,  Cambridge. 

Here  it  may  be  well  to  give  a  brief  account  of  that  Uni- 
versity, as  it  was  the  alma  mater  of  four  out  of  the  six  mathe- 
maticians discussed  in  this  course  of  lectures,  f 

At  that  time  the  University  of  Cambridge  consisted  of  seven- 
teen colleges,  each  of  which  had  an  independent  endowment, 
buildings,  master,  fellows  and  scholars.  The  endowments,  gen- 
erally in  the  shape  of  lands,  have  come  down  from  ancient  times; 
for  example,  Trinity  College  was  founded  by  Henry  VIII  in 

*  This  Lecture  was  delivered  April  12,  1901. — EDITORS. 

t  Dr.  Macfarlane's  first  course  included  the  first  six  lectures  given  in  this 
volume. — EDITORS. 

7 


8  TEN  BRITISH   MATHEMATICIANS 

1546,  and  at  the  beginning  of  the  i9th  century  it  consisted  of  a 
master,  60  fellows  and  72  scholars.  Each  college  was  provided 
with  residence  halls,  a  dining  hall,  and  a  chapel.  Each  college 
had  its  own  staff  of  instructors  called  tutors  or  lecturers,  and 
the  function  of  the  University  apart  from  the  colleges  was 
mainly  to  examine  for  degrees.  Examinations  for  degrees  con- 
sisted of  a  pass  examination  and  an  honors  examination,  the 
latter  called  a  tripos.  Thus,  the  mathematical  tripos  meant 
the  examinations  of  candidates  for  the  degree  of  Bachelor  of 
Arts  who  had  made  a  special  study  of  mathematics.  The 
examination  was  spread  over  a  week,  and  those  who  obtained 
honors  were  divided  into  three  classes,  the  highest  class  being 
called  wranglers,  and  the  highest  man  among  the  wranglers, 
senior  wrangler.  In  more  recent  times  this  examination  de- 
veloped into  what  De  Morgan  called  a  "  great  writing  race;" 
the  questions  being  of  the  nature  of  short  problems.  A  candidate 
put  himself  under  the  training  of  a  coach,  that  is,  a  mathema- 
tician who  made  it  a  business  to  study  the  kind  of  problems 
likely  to  be  set,  and  to  train  men  to  solve  and  write  out  the 
solution  of  as  many  as  possible  per  hour.  As  a  consequence 
the  lectures  of  the  University  professors  and  the  instruction  of 
the  college  tutors  were  neglected,  and  nothing  was  studied  ex- 
cept what  would  pay  in  the  tripos  examination.  Modifications 
have  been  introduced  to  counteract  these  evils,  and  the  con- 
ditions have  been  so  changed  that  there  are  now  no  senior 
wranglers.  The  tripos  examination  used  to  be  followed  almost 
immediately  by  another  examination  in  higher  mathematics  to 
determine  the  award  of  two  prizes  named  the  Smith's  prizes. 
"  Senior  wrangler  "  was  considered  the  greatest  academic  dis- 
tinction in  England. 

In  1812  Peacock  took  the  rank  of  second  wrangler,  and  the 
second  Smith's  prize,  the  senior  wrangler  being  John  Herschel. 
Two  years  later  he  became  a  candidate  for  a  fellowship  in  his 
college  and  won  it  immediately,  partly  by  means  of  his  exten- 
sive and  "accurate  knowledge  of  the  classics.  A  fellowship  then 
meant  about  £200  a  year,  tenable  for  seven  years  provided 
the  Fellow  did  not  marry  meanwhile,  and  capable  of  being 


GEORGE  PEACOCK  9 

extended  after  the  seven  years  provided  the  Fellow  took  clerical 
Orders.  The  limitation  to  seven  years,  although  the  Fellow 
devoted  himself  exclusively  to  science,  cut  short  and  prevented 
by,  anticipation  the  career  of  many  a  laborer  for  the  advance- 
ment of  science.  Sir  Isaac  Newton  was  a  Fellow  of  Trinity 
College,  and  its  limited  terms  nearly  deprived  the  world  of  the 
Principia. 

The  year  after  taking  a  Fellowship,  Peacock  was  appointed 
a  tutor  and  lecturer  of  his  college,  which  position  he  continued 
to  hold  for  many  years.  At  that  time  the  state  of  mathematical 
learning  at  Cambridge  was  discreditable.  How  could  that  be? 
you  may  ask;  was  not  Newton  a  professor  of  mathematics  in 
that  University?  did  he  not  write  the  Principia  in  Trinity 
College?  had  his  influence  died  out^so  soon?  The  true  reason 
was  he  was  worshipped  too  much  as  an  authority;  the  Univer- 
sity had  settled  dov/n  to  the  study  of  Newton  instead  of  Nature, 
and  they  had  followed  him  in  one  grand  mistake — the  ignoring 
of  the  differential  notation  in  the  calculus.  Students  of  the 
differential  calculus  are  more  or  less  familiar  with  the  controversy 
which  raged  over  the  respective  claims  of  Newton  and  Leibnitz 
to  the  invention  of  the  calculus;  rather  over  the  question  whether 
Leibnitz  was  an  independent  inventor,  or  appropriated  the 
fundamental  ideas  from  Newton's  writings  and  correspondence, 
merely  giving  them  a  new  clothing  in  the  form  of  the  differential 
notation.  Anyhow,  Newton's  countrymen  adopted  the  latter 
alternative;  they  clung  to  the  fluxional  notation  of  Newton; 
and  following  Newton,  they  ignored  the  notation  of  Leibniz 
and  everything  written  in  that  notation.  The  Newtonian 
notation  is  as  follows:  If  y  denotes  a  fluent,  then  y  denotes  its 
fluxion,  and  y  the  fluxion  of  y-y  \iy  itself  be  considered  a  fluxion, 
then  y'  denotes  its  fluent,  and  y"  the  fluent  of  y'  and  so  on; 
a  differential  is  denoted  by  o.  In  the  notation  of  Leibnitz  y 

is  written  -?-,  y  is  written  •—,  y'  is    j  ydx,  and  so  on.     The 

ax   '  ax*  J 

result  of  this  Chauvinism  on  the  part  of  the  British  mathema- 
ticians of  the  eighteenth  century  was  that  the  developments  of 
the  calculus  were  made  by  the  contemporary  mathematicians 


10  TEN   BRITISH  MATHEMATICIANS 

of  the  Continent,  namely,  the  Bernoullis,  Euler,  Clairault, 
Delambre,  Lagrange,  Laplace,  Legendre.  At  the  beginning 
of  the  i Qth  century,  there  was  only  one  mathematician  in  Great 
Britain  (namely  Ivory,  a  Scotsman)  who  was  familiar  with^the 
achievements  of  the  Continental  mathematicians.  Cambridge 
University  in  particular  was  wholly  given  over  not  merely  to  the 
use  of  the  fluxional  notation  but  to  ignoring  the  differential 
notation.  The  celebrated  saying  of  Jacobi  was  then  literally 
true,  although  it  had  ceased  to  be  true  when  he  gave  it  utterance. 
He  visited  Cambridge  about  1842.  When  dining  as  a  guest  at 
the  high  table  of  one  of  the  colleges  he  was  asked  who  in  his 
opinion  was  the  greatest  of  the  living  mathematicians  of  England ; 
his  reply  was  "  There  is  none." 

Peacock,  in  common  with  many  other  students  of  his  own 
standing,  was  profoundly  impressed  with  the  need  of  reform, 
and  while  still  an  undergraduate  formed  a  league  with  Babbage 
and  Herschel  to  adopt  measures  to  bring  it  about.  In  1815  they 
formed  what  they  called  the  Analytical  Society,  the  object  of 
which  was  stated  to  be  to  advocate  the  J'ism  of  the  Continent 
versus  the  dot-age  of  the  University.  Evidently  the  members 
of  the  new  society  were  armed  with  wit  as  well  as  mathematics. 
Of  these  three  reformers,  Babbage  afterwards  became  celebrated 
as  the  inventor  of  an  analytical  engine,  which  could  not  only 
perform  the  ordinary  processes  of  arithmetic,  but,  when  set 
with  the  proper  data,  could  tabulate  the  values  of  any  function 
and  print  the  results.  A  part  of  the  machine  was  constructed, 
but  the  inventor  and  the  Government  (which  was  supplying  the 
funds)  quarrelled,  in  consequence  of  which  the  complete  machine 
exists  only  in  the  form  of  drawings.  These  are  now  in  the 
possession  of  the  British  Government,  and  a  scientific  commis- 
sion appointed  to  examine  them  has  reported  that  the  engine 
could  be  constructed.  The  third  reformer — Herschel — was  a 
son  of  Sir  William  Herschel,  the  astronomer  who  discovered 
Uranus,  and  afterwards  as  Sir  John  Herschel  became  famous 
as  an  astronomer  and  scientific  philosopher. 

The  first  movement  on  the  part  of  the  Analytical  Society 
was  to  translate  from  the  French  the  smaller  work  of  Lacroix  on 


GEORGE  PEACOCK  11 

the  differential  and  integral  calculus;  it  was  published  in  1816. 
At  that  time  the  best  manuals,  as  well  as  the  greatest  works  on 
mathematics,  existed  in  the  French  language.  Peacock  followed 
up  the  translation  with  a  volume  containing  a  copious  Collection 
of  Examples  of  the  Application  of  the  Differential  and  Integral 
Calculus,  which  was  published  in  1820.  The  sale  of  both  books 
was  rapid,  and  contributed  materially  to  further  the  object  of 
the  Society.  Then  high  wranglers  of  one  year  became  the 
examiners  of  the  mathematical  tripos  three  or  four  years  after- 
wards. Peacock  was  appointed  an  examiner  in  1817,  and  he  did 
not  fail  to  make  use  of  the  position  as  a  powerful  lever  to  advance 
the  cause  of  reform.  In  his  questions  set  for  the  examination 
the  differential  notation  was  for  the  first  time  officially  employed 
in  Cambridge.  The  innovation  did  not  escape  censure,  but  he 
wrote  to  a  friend  as  follows:  "  I  assure  you  that  I  shall  never 
cease  to  exert  myself  to  the  utmost  in  the  cause  of  reform,  and 
that  I  will  never  decline  any  office  which  may  increase  my 
power  to  effect  it.  I  am  nearly  certain  of  being  nominated  to 
the  office  of  Moderator  in  the  year  1818-1819,  and  as  I  am  an 
examiner  in  virtue  of  my  office,  for  the  next  year  I  shall  pursue 
a  course  even  more  decided  than  hitherto,  since  I  shall  feel  that 
men  have  been  prepared  for  the  change,  and  will  then  be 
enabled  to  have  acquired  a  better  system  by  the  publication  of 
improved  elementary  books.  I  have  considerable  influence  as 
a  lecturer,  and  I  will  not  neglect  it.  It  is  by  silent  perseverance 
only,  that  we  can  hope  to  reduce  the  many-headed  monster  of 
prejudice  and  make  the  University  answer  her  character  as  the 
loving  mother  of  good  learning  and  science."  These  few  sen- 
tences give  an  insight  into  the  character  of  Peacock:  he  was 
an  ardent  reformer  and  a  few  years  brought  success  to  the  cause 
of  the  Analytical  Society. 

Another  reform  at  which  Peacock  labored  was  the  teaching 
of  algebra.  In  1830  he  published  a  Treatise  on  Algebra  which 
had  for  its  object  the  placing  of  algebra  on  a  true  scientific 
basis,  adequate  for  the  development  which  it  had  received  at 
the  hands  of  the  Continental  mathematicians.  As  to  the  state 
of  the  science  of  algebra  in  Great  Britain,  it  may  be  judged 


12  TEN   BRITISH   MATHEMATICIANS 

of  by  the  following  facts.  Baron  Maseres,  a  Fellow  of  Clare 
College,  Cambridge,  and  William  Frend,  a  second  wrangler, 
had  both  written  books  protesting  against  the  use  of  the  nega- 
tive quantity.  Frend  published  his  Principles  of  A Igebra  in  1 796 > 
and  the  preface  reads  as  follows :  "  The  ideas  of  number  are  the 
clearest  and  most  distinct  of  the  human  mind;  the  acts  of  the 
mind  upon  them  are  equally  simple  and  clear.  There  cannot 
be  confusion  in  them,  unless  numbers  too  great  for  the  com- 
prehension of  the  learner  are  employed,  or  some  arts  are  used 
which  are  not  justifiable.  The  first  error  in  teaching  the  first 
principles  of  algebra  is  obvious  on  perusing  a  few  pages  only 
of  the  first  part  of  Maclaurin's  Algebra.  Numbers  are  there 
divided  into  two  sorts,  positive  and  negative;  and  an  attempt 
is  made  to  explain  the  nature  of  negative  numbers  by  allusion 
to  book  debts  and  other  arts.  Now  when  a  person  cannot 
explain  the  principles  of  a  science  without  reference  to  a  meta- 
phor, the  probability  is,  that  he  has  never  thought  accurately 
upon  the  subject.  A  number  may  be  greater  or  less  than  another 
number;  it  may  be  added  to,  taken  from,  multiplied  into,  or 
divided  by,  another  number;  but  in  other  respects  it  is  very 
intractable;  though  the  whole  world  should  be  destroyed,  one 
will  be  one,  and  three  will  be  three,  and  no  art  whatever  can 
change  their  nature.  You  may  put  a  mark  before  one,  which 
it  will  obey;  it  submits  to  be  taken  away  from  a  number  greater 
than  itself,  but  to  attempt  to  take  it  away  from  a  number  less 
than  itself  is  ridiculous.  Yet  this  is  attempted  by  algebraists 
who  talk  of  a  number  less  than  nothing;  of  multiplying  a  negative 
number  into  a  negative  number  and  thus  producing  a  positive 
number;  of  a  number  being  imaginary.  Hence  they  talk  of 
two  roots Jx>  every  equation  of  the  second  order,  and  the  learner 
is  to  try  which  will  succeed  in  a  given  equation;  they  talk  of 
solving  an  equation  which  requires  two  impossible  roots  to  make 
it  soluble;  they  can  find  out  some  impossible  numbers  which 
being  multiplied  together  produce  unity.  This  is  all  jargon, 
at  which  common  sense  recoils;  but  from  its  having  been  once 
adopted,  like  many  other  figments,  it  finds  the  most  strenuous 
supporters  among  those  who^love  to  take  things  upon  trust  and 


GEORGE   PEACOCK  13 

hate  the  colour  of  a  serious  thought."  So  far,  Frend.  Peacock 
knew  that  Argand,  Francais  and  Warren  had  given  what 
seemed  to  be  an  explanation  not  only  of  the  negative  quantity 
but  of  the  imaginary,  and  his  object  was  to  reform  the  teaching 
of  algebra  so  as  to  give  it  a  true  scientific  basis. 

At  that  time  every  part  of  exact  science  was  languishing  in 
Great  Britain.  Here  is  the  description  given  by  Sir  John 
Herschel:  "The  end  of  the  i8th  and  the  beginning  of  the  igth 
century  were  remarkable  for  the  small  amount  of  scientific  move- 
ment going  on  in  Great  Britain,  especially  in  its  more  exact 
departments.  Mathematics  were  at  the  last  gasp,  and  Astronomy 
nearly  so — I  mean  in  those  members  of  its  frame  which  depend 
upon  precise  measurement  and  systematic  calculation.  The 
chilling  torpor  of  routine  had  begun  to  spread  itself  over  all 
those  branches  of  Science  which  wanted  the  excitement  of 
experimental  research."  To  elevate  astronomical  science  the 
Astronomical  Society  of  London  was  founded,  and  our  three 
reformers  Peacock,  Babbage  and  Herschel  were  prime  movers 
in  the  undertaking.  Peacock  was  one  of  the  most  zealous 
promoters  of  an  astronomical  observatory  at  Cambridge,  and 
one  of  the  founders  of  the  Philosophical  Society  of  Cambridge. 

The  year  1831  saw  the  beginning  of  one  of  the  greatest 
scientific  organizations  of  modern  times.  That  year  the  British 
Association  for  the  Advancement  of  Science  (prototype  of  the 
American,  French  and  Australasian  Associations)  held  its  first 
meeting  in  the  ancient  city  of  York.  Its  objects  were  stated 
to  be:  first,  to  give  a  stronger  impulse  and  a  more  systematic 
direction  to  scientific  enquiry;  second,  to  promote  the  inter- 
course of  those  who  cultivate  science  in  different  parts  of  the 
British  Empire  with  one  another  and  with  foreign  philosophers; 
third,  to  obtain  a  more  general  attention  to  the  objects  of 
science,  and  the  removal  of  any  disadvantages  of  a  public  kind 
which  impede  its  progress.  One  of  the  first  resolutions  adopted 
was  to  procure  reports  on  the  state  and  progress  of  particular 
sciences,  to  be  drawn  up  from  time  to  time  by  competent  per- 
sons for  the  information  of  the  annual  meetings,  and  the  first 
to  be  placed  on  the  list  was  a  report  on  the  progress  of  mathe- 


14  TEN   BRITISH   MATHEMATICIANS 

matical  science.  Dr.  Whewell,  the  mathematician  and  phil- 
osopher, was  a  Vice-president  of  the  meeting:  he  was  instructed 
to  select  the  reporter.  He  first  asked  Sir  W.  R.  Hamilton,  who 
declined;  he  then  asked  Peacock,  who  accepted.  Peacock  had 
his  report  ready  for  the  third  meeting  of  the  Association,  which 
was  held  in  Cambridge  in  1833;  although  limited  to  Algebra, 
Trigonometry,  and  the  Arithmetic  of  Sines,  it  is  one  of  the  best 
of  the  long  series  of  valuable  reports  which  have  been  prepared 
for  and  printed  by  the  Association. 

In  1837  he  was  appointed  Lowndean  professor  of  astronomy 
in  the  University  of  Cambridge,  the  chair  afterwards  occupied 
by  Adams,  the  co-discoverer  of  Neptune,  and  now  occupied  by 
Sir  Robert  Ball,  celebrated  for  his  Theory  of  Screws.  In  1839 
he  was  appointed  Dean  of  Ely,  the  diocese  of  Cambridge.  While 
holding  this  position  he  wrote  a  text  book  on  algebra  in  two 
volumes,  the  one  called  Arithmetical  Algebra,  and  the  other 
Symbolical  Algebra.  Another  object  of  reform  was  the  stat- 
utes of  the  University;  he  worked  hard  at  it  and  was  made 
a  member  of  a  commission  appointed  by  the  Government  for 
the  purpose;  but  he  died  on  November  8,  1858,  in  the  68th  year 
of  his  age.  His  last  public  act  was  to  attend  a  meeting  of  the 
Commission. 

Peacock's  main  contribution  to  mathematical  analysis  is  his 
attempt  to  place  algebra  on  a  strictly  logical  basis.  He  founded 
what  has  been  called  the  philological  or  symbolical  school  of 
mathematicians;  to  which  Gregory,  De  Morgan  and  Boole 
belonged.  His  answer  to  Maseres  and  Frend  was  that  the 
science  of  algebra  consisted  of  two  parts — arithmetical  algebra 
and  symbolical  algebra — and  that  they  erred  in  restricting  the 
science  to  the  arithmetical  part.  His  view  of  arithmetical 
algebra  is  as  follows:  "  In  arithmetical  algebra  we  consider 
symbols  as  representing  numbers,  and  the  operations  to  which 
they  are  submitted  as  included  in  the  same  definitions  as  in 
common  arithmetic;  the  signs  +  and  —  denote  the  operations 
of  addition  and  subtraction  in  their  ordinary  meaning  only, 
and  those  operations  are  considered  as  impossible  in  all  cases 
where  the  symbols  subjected  to  them  possess  values  which 


GEORGE  PEACOCK  15 

would  render  them  so  in  case  they  were  replaced  by  digital 
numbers;  thus  in  expressions  such  as  a -{-b  we  must  suppose 
a  and  b  to  be  quantities  of  the  same  kind;  in  others,  like  a  —  b, 
we  must  suppose  a  greater  than  b  and  therefore  homogeneous 

with  it;  in  products  and  quotients,  like  ab  and  -  we  must  suppose 

the  multiplier  and  divisor  to  be  abstract  numbers;  all  results 
whatsoever,  including  negative  quantities,  which  are  not  strictly 
deducible  as  legitimate  conclusions  from  the  definitions  of  the 
several  operations  must  be  rejected  as  impossible,  or  as  foreign 
to  the  science." 

Peacock's  principle  may  be  stated  thus:  the  elementary 
symbol  of  arithmetical  algebra  denotes  a  digital,  i.e.,  an  integer 
number;  and  every  combination  of  elementary  symbols  must 
reduce  to  a  digital  number,  otherwise  it  is  impossible  or  foreign 
to  the  science.  If  a  and  b  are  numbers,  then  a -\-b  is  always 
a  number;  but  a— b  is  a  number  only  when  b  is  less  than  a. 
Again,  under  the  same  conditions,  ab  is  always  a  number,  but 

-  is  really  a  number  only  when  b  is  an  exact  divisor  of  a. 
b 

Hence  we  are  reduced  to  the  following  dilemma:  Either  -  must 

b 

be  held  to  be  an  impossible  expression  in  general,  or  else  the 
meaning  of  the  fundamental  symbol  of  algebra  must  be  extended 
so  as  to  include  rational  fractions.  If  the  former  horn  of  the 
dilemma  is  chosen,  arithmetical  algebra  becomes  a  mere  shadow; 
if  the  latter  horn  is  chosen,  the  operations  of  algebra  cannot 
be  defined  on  the  supposition  that  the  elementary  symbol  is  an 
integer  number.  Peacock  attempts  to  get  out  of  the  difficulty 
by  supposing  that  a  symbol  which  is  used  as  a  multiplier  is 
always  an  integer  number,  but  that  a  symbol  in  the  place  of  the 
multiplicand  may  be  a  fraction.  For  instance,  in  ab,  a  can  denote 
only  an  integer  number,  but  b  may  denote  a  rational  fraction. 
Now  there  is  no  more  fundamental  principle  in  arithmetical 
algebra  than  that  ab  =  ba;  which  would  be  illegitimate  on 
Peacock's  principle. 

One  of  the  earliest  English  writers  on  arithmetic  is  Robert 


16  TEN   BRITISH   MATHEMATICIANS 

Record,  who  dedicated  his  work  to  King  Edward  the  Sixth. 
The  author  gives  his  treatise  the  form  of  a  dialogue  between 
master  and  scholar.  The  scholar  battles  long  over  this  diffi- 
culty,— that  multiplying  a  thing  could  make  it  less.  The  master 
attempts  to  explain  the  anomaly  by  reference  to  proportion; 
that  the  product  due  to  a  fraction  bears  the  same  proportion  to 
the  thing  multiplied  that  the  fraction  bears  to  unity.  But  the 
scholar  is  not  satisfied  and  the  master  goes  on  to  say:  "  If  I 
multiply  by  more  than  one,  the  thing  is  increased;  if  I  take  it 
but  once,  it  is  not  changed,  and  if  I  take  it  less  than  once,  it 
cannot  be  so  much  as  it  was  before.  Then  seeing  that  a  fraction 
is  less  than  one,  if  I  multiply  by  a  fraction,  it  follows  that  I  do 
take  it  less  than  once."  Whereupon  the  scholar  replies,  "  Sir, 
I  do  thank  you  much  for  this  reason, — and  I  trust  that  I  do  per- 
ceive the  thing." 

The  fact  is  that  even  in  arithmetic  the  two  processes  of 
multiplication  and  division  are  generalized  into  a  common  mul- 
tiplication ;  and  the  difficulty  consists  in  passing  from  the  origi- 
nal idea  of  multiplication  to  the  generalized  idea  of  a  tensor, 
which  idea  includes  compressing  the  magnitude  as  well  as 
stretching  it.  Let  m  denote  an  integer  number;  the  next  step 

is  to  gain  the  idea  of  the  reciprocal  of  m,  not  as  —  but  simply  as 

m 

/m.  When  m  and  /n  are  compounded  we  get  the  idea  of  a 
rational  fraction;  for  in  general  m/n  will  not  reduce  to  a  number 
nor  to  the  reciprocal  of  a  number. 

Suppose,  however,  that  we  pass  over  this  objection;  how 
does  Peacock  lay  the  foundation  for  general  algebra?  He  calls 
it  symbolical  algebra,  and  he  passes  from  arithmetical  algebra 
to  symbolical  algebra  in  the  following  manner:  "  Symbolical 
algebra  adopts  the  rules  of  arithmetical  algebra  but  removes 
altogether  their  restrictions;  thus  symbolical  subtraction  dif- 
fers from  the  same  operation  in  arithmetical  algebra  in  being 
possible  for  all  relations  of  value  of  the  symbols  or  expressions 
employed.  All  the  results  of  arithmetical  algebra  which  are 
deduced  by  the  application  of  its  rules,  and  which  are  general 
in  form  though  particular-  in  value,  are  results  likewise  of 


GEORGE  PEACOCK  17 

symbolical  algebra  where  they  are  general  in  value  as  well  as  in 
form;  thus  the  product  of  am  and  a"  which  is  am+n  when  m 
and  n  are  whole  numbers  and  therefore  general  in  form  though 
particular  in  value,  will  be  their  product  likewise  when  m  and 
n  are  general  in  value  as  well  as  in  form;  the  series  for  (a+b)n 
determined  by  the  principles  of  arithmetical  algebra  when  n 
is  any  whole  number,  if  it  be  exhibited  in  a  general  form,  without 
reference  to  a  final  term,  may  be  shown  upon  the  same  principle 
to  the  equivalent  series  for  (a-\-b}n  when  n  is  general  both  in 
form  and  value." 

The  principle  here  indicated  by  means  of  examples  was 
named  by  Peacock  the  "  principle  of  the  permanence  of  equiva- 
lent forms,"  and  at  page  59  of  the  Symbolical  Algebra  it  is  thus 
enunciated:  "  Whatever  algebraical  forms  are  equivalent  when 
the  symbols  are  general  in  form,  but  specific  in  value,  will  be 
equivalent  likewise  when  the  symbols  are  general  in  value  as 
well  as  in  form." 

For  example,  let  a,  b,  c,  d  denote  any  integer  numbers,  but 
subject  to  the  restrictions  that  b  is  less  than  a,  and  d  less  than 
c;  it  may  then  be  shown  arithmetically  that 

(a  —  b}(c  — d)  =  ac-\-bd— ad— be. 

Peacock's  principle  says  that  the  form  on  the  left  side  is  equiva- 
lent to  the  form  on  the  right  side,  not  only  when  the  said 
restrictions  of  being  less  are  removed,  but  when  a,  b,  c,  d  denote 
the  most  general  algebraical  symbol.  It  means  that  a,  b,  c,  d 
may  be  rational  fractions,  or  surds,  or  imaginary  quantities, 

or  indeed  operators  such  as  — .  The  equivalence  is  not  estab- 
lished by  means  of  the  nature  of  the  quantity  denoted;  the 
equivalence  is  assumed  to  be  true,  and  then  it  is  attempted 
to  find  the  different  interpretations  which  may  be  put  on  the 
symbol. 

It  is  not  difficult  to  see  that  the  problem  before  us 
involves  the  fundamental  problem  of  a  rational  logic  or 
theory  of  knowledge;  namely,  how  are  we  able  to  ascend  from 
particular  truths  to  more  general  truths.  If  a,  b,  c,  d  denote 


18  TEN  BRITISH  MATHEMATICIANS 

integer  numbers,  of  which  b  is  less  than  a  and  d  less  than  c, 
then 

(a  —  b}  (c—d)=ac-\-bd—ad—bc. 

It  is  first  seen  that  the  above  restrictions  may  be  removed, 
and  still  the  above  equation  hold.  But  the  antecedent  is  still 
too  narrow;  the  true  scientific  problem  consists  in  specifying 
the  meaning  of  the  symbols,  which,  and  only  which,  will  admit 
of  the  forms  being  equal.  It  is  not  to  find  some  meanings,  but 
the  most  general  meaning,  which  allows  the  equivalence  to  be 
true.  Let  us  examine  some  other  cases;  we  shall  find  that 
Peacock's  principle  is  not  a  solution  of  the  difficulty;  the  great 
logical  process  of  generalization  cannot  be  reduced  to  any  such 
easy  and  arbitrary  procedure.  When  a,  m,  n  denote  integer 
numbers,  it  can  be  shown  that 


According  to  Peacock  the  form  on  the  left  is  always  to  be  equal  to 
the  form  on  the  right,  and  the  meanings  of  a,  m,  n  are  to  be  found 
by  interpretation.  Suppose  that  a  takes  the  form  of  the  incom- 
mensurate quantity  e,  the  base  of  the  natural  system  of  logar- 
ithms. A  number  is  a  degraded  form  of  a  complex  quantity 
(p-\-q^  —  i)  and  a  complex  quantity  is  a  degraded  form  of  a 
quaternion;  consequently  one  meaning  which  may  be  assigned 
to  m  and  n  is  that  of  quaternion.  Peacock's  principle  would 
lead  us  to  suppose  that  emen  =  em+n,  m  and  n  denoting  qua- 
ternions; but  that  is  just  what  Hamilton,  the  inventor  of  the 
quaternion  generalization,  denies.  There  are  reasons  for  believ- 
ing that  he  was  mistaken,  and  that  the  forms  remain  equivalent 
even  under  that  extreme  generalization  of  m  and  n;  but  the 
point  is  this  :  it  is  not  a  question  of  conventional  definition  and 
formal  truth;  it  is  a  question  of  objective  definition  and  real 
truth.  Let  the  symbols  have  the  prescribed  meaning,  does  or 
does  not  the  equivalence  still  hold?  And  if  it  does  not  hold, 
what  is  the  higher  or  more  complex  form  which  the  equivalence 
assumes? 


AUGUSTUS  DE  MORGAN* 

/ 
(1806-1871) 

AUGUSTUS  DE  MORGAN  was  born  in  the  month  of  June  at 
Madura  in  the  presidency  of  Madras,  India;  and  the  year  of 
his  birth  may  be  found  by  solving  a  conundrum  proposed  by 
himself,  "  I  was  x  years  of  age  in  the  year  #2."  The  problem 
is  indeterminate,  but  it  is  made  strictly  determinate  by  the 
century  of  its  utterance  and  the  limit  to  a  man's  life.  His  father 
was  Col.  De  Morgan,  who  held  various  appointments  in  the  service 
of  the  East  India  Company.  His  mother  was  descended  from 
James  Dodson,  who  computed  a  table  of  anti-logarithms,  that  is, 
the  numbers  corresponding  to  exact  logarithms.  It  was  the  time 
of  the  Sepoy  rebellion  in  India,  and  Col.  De  Morgan  removed 
his  family  to  England  when  Augustus  was  seven  months  old. 
As  his  father  and  grandfather  had  both  been  born  in  India, 
De  Morgan  used  to  say  that  he  was  neither  English,  nor  Scottish, 
nor  Irish,  but  a  Briton  "  unattached,"  using  the  technical 
term  applied  to  an  undergraduate  of  Oxf  >rd  or  Cambridge  who 
is  not  a  member  of  any  one  of  the  Colleges. 

When^De  Morgan  was  ten  years  old,  his  father  died.  Mrs. 
De  Morgan  resided  at  various  places  in  the  southwest  of  England, 
and  her  son  received  his  elementary  education  at  various  schools 
of  no  great  account.  His  mathematical  talents  were  unnoticed 
till  he  had  reached  the  age  of  fourteen.  A  friend  of  the  family 
accidentally  discovered  him  making  an  elaborate  drawing  of 
a  figure  in  Euclid  with  ruler  and  compasses,  and  explained  to 
him  the  aim  of  Euclid,  and  gave  him  an  initiation  into  demon- 
stration. 

De  Morgan  suffered  from  a  physical  defect — one  of  his  eyes 
was  rudimentary  and  useless.  As  a  consequence,  he  did  not 

*  This  Lecture  was  delivered  April  13,  1901. — EDITORS. 
19 


20  TEN   BRITISH   MATHEMATICIANS 

join  in  the  sports  of  the  other  boys,  and  he  was  even  made  the 
victim  of  cruel  practical  jokes  by  some  schoolfellows.  Some 
psychologists  have  held  that  the  perception  of  distance  and  of 
solidity  depends  on  the  action  of  two  eyes,  but  De  Morgan 
testified  that  so  far  as  he  could  make  out  he  perceived  with  his 
one  eye  distance  and  solidity  just  like  other  people. 

He  received  his  secondary  education  from  Mr.  Parsons, 
a  Fellow  of  Oriel  College,  Oxford,  who  could  appreciate 
classics  much  better  than  mathematics.  His  mother  was  an 
active  and  ardent  member  of  the  Church  of  England,  and 
desired  that  her  son  should  become  a  clergyman;  but  by  this 
time  De  Morgan  had  begun  to  show  his  non-grooving  dispo- 
sition, due  no  doubt  to  some  extent  to  his  physical  infirmity. 
At  the  age  of  sixteen  he  was  entered  at  Trinity  College,  Cam- 
bridge, where  he  immediately  came  under  the  tutorial  influence 
of  Peacock  and  Whewell.  They  became  his  life-long  friends; 
from  the  former  he  derived  an  interest  in  the  renovation  of 
algebra,  and  from  the  latter  an  interest  in  the  renovation  of 
logic — the  two  subjects  of  his  future  life  work. 

At  college  the  flute,  on  which  he  played  exquisitely,  was  his 
recreation.  He  took  no  part  in  athletics  but  was  prominent 
in  the  musical  clubs.  His  love  of  knowledge  for  its  own  sake 
interfered  with  training  for  the  great  mathematical  race;  as 
a  consequence  he  came  out  fourth  wrangler.  This  entitled  him 
to  the  degree  of  Bachelor  of  Arts;  but  to  take  the  higher  degree 
of  Master  of  Arts  and  thereby  become  eligible  for  a  fellowship 
it  was  then  necessary  to  pass  a  theological  test.  To  the  sign- 
ing of  any  such  test  De  Morgan  felt  a  strong  objection,  although 
he  had  been  brought  up  in  the  Church  of  England.  About 
1875  theological  tests  for  academic  degrees  were  abolished  in 
the  Universities  of  Oxford  and  Cambridge. 

As  no  career  was  open  to  him  at  his  own  university,  he 
decided  to  go  to  the  Bar,  and  took  up  residence  in  London; 
but  he  much  preferred  teaching  mathematics  to  reading  law. 
About  this  time  the  movement  for  founding  the  London  Uni- 
versity took  shape.  The  two  ancient  universities  were  so 
guarded  by  theological  tests  that  no  Jew  or  Dissenter  from  the 


AUGUSTUS  DE   MORGAN  21 

Church  of  England  could  enter  as  a  student;  still  less  be 
appointed  to  any  office.  A  body  of  liberal-minded  men  resolved 
to  meet  the  difficulty  by  establishing  in  London  a  University 
on  the  principle  of  religious  neutrality.  De  Morgan,  then  22 
years  of  age,  was  appointed  Professor  of  Mathematics.  His 
introductory  lecture  "  On  the  study  of  mathematics  "  is  a  dis- 
course upon  mental  education  of  permanent  value  which  has 
been  recently  reprinted  in  the  United  States. 

The  London  University  was  a  new  institution,  and  the 
relations  of  the  Council  of  management,  the  Senate  of  professors 
and  the  body  of  students  were  not  well  defined.  A  dispute 
arose  between  the  professor  of  anatomy  and  his  students,  and  in 
consequence  of  the  action  taken  by  the  Council,  several  of  the 
professors  resigned,  headed  by  De  Morgan.  Another  professor 
of  mathematics  was  appointed,  who  was  accidentally  drowned 
a  few  years  later.  De  Morgan  had  shown  himself  a  prince  of 
teachers:  he  was  invited  to  return  to  his  chair,  which  thereafter 
became  the  continuous  center  of  his  labors  for  thirty  years. 

The  same  body  of  reformers — headed  by  Lord  Brougham, 
a  Scotsman  eminent  both  in  science  and  politics — who  had 
instituted  the  London  University,  founded  about  the  same  time 
a  Society  for  the  Diffusion  of  Useful  Knowledge.  Its  object 
was  to  spread  scientific  and  other  knowledge  by  means  of  cheap 
and  clearly  written  treatises  by  the  best  writers  of  the  time. 
One  of  its  most  voluminous  and  effective  writers  was  De  Morgan. 
He  wrote  a  great  work  on  The  Differential  and  Integral  Calculus 
which  was  published  by  the  Society;  and  he  wrote  one-sixth  of 
the  articles  in  the  Penny  Cyclopedia,  published  by  the  Society, 
and  issued  in  penny  numbers.  When  De  Morgan  came  to  reside 
in  London  he  found  a  congenial  friend  in  William  Frend,  not- 
withstanding his  mathematical  heresy  about  negative  quan- 
tities. Both  were  arithmeticians  and  actuaries,  and  their 
religious  views  were  somewhat  similar.  Frend  lived  in  what 
was  then  a  suburb  of  London,  in  a  country-house  formerly 
occupied  by  Daniel  Defoe  and  Isaac  Watts.  De  Morgan  with 
his  flute  was  a  welcome  visitor;  and  in  1837  he  married  Sophia 
Elizabeth,  one  of  Frend's  daughters. 


22  TEN   BRITISH   MATHEMATICIANS 

The  London  University  of  which  De  Morgan  was  a  pro- 
fessor was  a  different  institution  from  the  University  of  London. 
The  University  of  London  was  founded  about  ten  years  later 
by  the  Government  for  the  purpose  of  granting  degrees  after 
examination,  without  any  qualification  as  to  residence.  The 
London  University  was  affiliated  as  a  teaching  college  with  the 
University  of  London,  and  its  name  was  changed  to  University 
College.  The  University  of  London  was  not  a  success  as  an 
examining  body;  a  teaching  University  was  demanded.  De 
Morgan  was  a  highly  successful  teacher  of  mathematics.  It  was 
his  plan  to  lecture  for  an  hour,  and  at  the  close  of  each  lecture 
to  give  out  a  number  of  problems  and  examples  illustrative 
of  the  subject  lectured  on;  his  students  were  required  to  sit 
down  to  them  and  bring  him  the  results,  which  he  looked  over 
and  returned  revised  before  the  next  lecture.  In  De  Morgan's 
opinion,  a  thorough  comprehension  and  mental  assimilation  of 
great  principles  far  outweighed  in  importance  any  merely 
analytical  dexterity  in  the  application  of  half-understood  prin- 
ciples to  particular  cases. 

De  Morgan  had  a  son  George,  who  acquired  great  distinction 
in  mathematics  both  at  University  College  and  the  University 
of  London.  He  and  another  like-minded  alumnus  conceived 
the  idea  of  founding  a  Mathematical  Society  in  London,  where 
mathematical  papers  would  be  not  only  received  (as  by  the 
Royal  Society)  but  actually  read  and  discussed.  The  first 
meeting  was  held  in  University  College;  De  Morgan  was  the 
first  president,  his  son  the  first  secretary.  It  was  the  beginning 
of  the  London  Mathematical  Society.  In  the  year  1866  the 
chair  of  mental  philosophy  in  University  College  fell  vacant. 
Dr.  Martineau,  a  Unitarian  clergyman  and  professor  of  mental 
philosophy,  was  recommended  formally  by  the  Senate  to  the 
Council;  but  in  the  Council  there  were  some  who  objected  to 
a  Unitarian  clergyman,  and  others  who  objected  to  theistic 
philosophy.  A  layman  of  the  school  of  Bain  and  Spencer  was 
appointed.  De  Morgan  considered  that  the  old  standard  of 
religious  neutrality  had  been  hauled  down,  and  forthwith 
resigned.  He  was  now  60  years  of  age.  His  pupils  secured  a 


AUGUSTUS   DE   MORGAN  23 

pension  of  $500  for  him,  but  misfortunes  followed.  Two  years 
later  his  son  George — the  younger  Bernoulli,  as  he  loved  to 
hear  him  called,  in  allusion  to  the  two  eminent  mathematicians 
of  that  name,  related  as  father  and  son — died.  This  blow  was 
followed  by  the  death  of  a  daughter.  Five  years  after  his  resig- 
nation from  University  College  De  Morgan  died  of  nervous 
prostration  on  March  18,  1871,  in  the  6$th  year  of  his  age. 

De  Morgan  was  a  brilliant  and  witty  writer,  whether  as  a 
controversialist  or  as  a  correspondent.  In  his  time  there  flour- 
ished two  Sir  William  Hamiltons  who  have  often  been  con- 
founded. The  one  Sir  William  was  a  baronet  (that  is,  inherited 
the  title),  a  Scotsman,  professor  of  logic  and  metaphysics  in  the 
University  of  Edinburgh;  the  other  was  a  knight  (that  is,  won 
the  title),  an  Irishman,  professor  of  astronomy  in  the  University 
of  Dublin.  The  baronet  contributed  to  logic  the  doctrine  of  the 
quantification  of  the  predicate;  the  knight,  whose  full  name 
was  William  Rowan  Hamilton,  contributed  to  mathematics  the 
geometric  algebra  called  Quaternions.  De  Morgan  was  inter- 
ested in  the  work  of  both,  and  corresponded  with  both;  but  the 
correspondence  with  the  Scotsman  ended  in  a  public  controversy, 
whereas  that  with  the  Irishman  was  marked  by  friendship  and 
terminated  only  by  death.  In  one  of  his  letters  to  Rowan, 
De  Morgan  says,  "Be  it  known  unto  you  that  I  have  discovered 
that  you  and  the  other  Sir  W.  H.  are  reciprocal  polars  with 
respect  to  me  (intellectually  and  morally,  for  the  Scottish 
baronet  is  a  polar  bear,  and  you,  I  was  going  to  say,  are  a  polar 
gentleman).  When  I  send  a  bit  of  investigation  to  Edinburgh, 
the  W.  H.  of  that  ilk  says  I  took  it  from  him.  When  I  send 
you  one,  you  take  it  from  me,  generalize  it  at  a  glance,  bestow 
it  thus  generalized  upon  society  at  large,  and  make  me  the  second 
discoverer  of  a  known  theorem." 

The  correspondence  of  De  Morgan  with  Hamilton  the  mathe- 
matician extended  over  twenty-four  years;  it  contains  discus- 
sions not  only  of  mathematical  matters,  but  also  of  subjects 
of  general  interest.  It  is  marked  by  geniality  on  the  part  of 
Hamilton  and  by  wit  on  the  part  of  De  Morgan.  The  following 
is  a  specimen:  Hamilton  wrote,  "  My  copy  of  Berkeley's  work 


24  TEN   BRITISH   MATHEMATICIANS 

is  not  mine;  like  Berkeley,  you  know,  I  am  an  Irishman."  De 
Morgan  replied,  "  Your  phrase  '  my  copy  is  not  mine  '  is  not 
a  bull.  It  is  perfectly  good  English  to  use  the  same  word  in 
two  different  senses  in  one  sentence,  particularly  when  there  is 
usage.  Incongruity  of  language  is  no  bull,  for  it  expresses  mean- 
ing. But  incongruity  of  ideas  (as  in  the  case  of  the  Irishman 
who  was  pulling  up  the  rope,  and  finding  it  did  not  finish,  cried 
out  that  somebody  had  cut  off  the  other  end  of  it)  is  the  genuine 
bull." 

De  Morgan  was  full  of  personal  peculiarities.  We  have 
noticed  his  almost  morbid  attitude  towards  religion,  and  the 
readiness  with  which  he  would  resign  an  office.  On  the  occasion 
of  the  installation  of  his  friend,  Lord  Brougham,  as  Rector  of 
the  University  of  Edinburgh,  the  Senate  offered  to  confer  on 
him  the  honorary  degree  of  LL.D.;  he  declined  the  honor  as 
a  misnomer.  He  once  printed  his  name:  Augustus  De  Morgan, 

H-O-M-O  •  P-A-U-C-A-R-U-M  •  L-I-T-E-R-A-R-U-M. 

He  disliked  the  country,  and  while  his  family  enjoyed  the  sea- 
side, and  men  of  science  were  having  a  good  time  at  a  meeting 
of  the  British  Association  in  the  country  he  remained  in  the  hot 
and  dusty  libraries  of  the  metropolis.  He  said  that  he  felt 
like  Socrates,  who  declared  that  the  farther  he  got  from  Athens 
the  farther  was  he  from  happiness.  He  never  sought  to  become 
a  Fellow  of  the  Royal  Society,  and  he  never  attended  a  meeting 
of  the  Society;  he  said  that  he  had  no  ideas  or  sympathies  in 
common  with  the  physical  philosopher.  His  attitude  was 
doubtless  due  to  his  physical  infirmity,  which  prevented  him 
from  being  either  an  observer  or  an  experimenter.  He  never 
voted  at  an  election,  and  he  never  visited  the  House  of  Commons, 
or  the  Tower,  or  Westminster  Abbey. 

Were  the  writings  of  De  Morgan  published  in  the  form  of 
collected  works,  they  would  form  a  small  library.  We  have 
noticed  his  writings  for  the  Useful  Knowledge  Society.  Mainly 
through  the  efforts  of  Peacock  and  Whewell,  a  Philosophical 
Society  had  been  inaugurated  at  Cambridge;  and  to  its  Trans- 
actions De  Morgan  contributed  four  memoirs  on  the  foundations 


AUGUSTUS  DE   MORGAN  25 

of  algebra,  and  an  equal  number  on  formal  logic.  The  best 
presentation  of  his  view  of  algebra  is  found  in  a  volume,  entitled 
Trigonometry  and  Double  Algebra,  published  in  1849;  and  his 
earlier  view  of  formal  logic  is  found  in  a  volume  published  in 
1847.  His  most  unique  work  is  styled  a  Budget  of  Paradoxes', 
it  originally  appeared  as  letters  in  the  columns  of  the  Athenaum 
journal;  it  was  revised  and  extended  by  De  Morgan  in  the  last 
years  of  his  life,  and  was  published  posthumously  by  his  widow. 
"  If  you  wish  to  read  something  entertaining,"  said  Professor 
Tait  to  me,  "  get  De  Morgan's  Budget  of  Paradoxes  out  of  the 
library."  We  shall  consider  more  at  length  his  theory  of 
algebra,  his  contribution  to  exact  logic,  and  his  Budget  of 
Paradoxes. 

In  my  last  lecture  I  explained  Peacock's  theory  of  algebra. 
It  was  much  improved  by  D.  F.  Gregory,  a  younger  member 
of  the  Cambridge  School,  who  laid  stress  not  on  the  permanence 
of  equivalent  forms,  but  on  the  permanence  of  certain  formal 
laws.  This  new  theory  of  algebra  as  the  science  of  symbols 
and  of  their  laws  of  combination  was  carried  to  its  logical  issue 
by  De  Morgan;  and  his  doctrine  on  the  subject  is  still  followed 
by  English  algebraists  in  general.  Thus  Chrystal  founds  his 
Textbook  of  Algebra  on  De  Morgan's  theory;  although  an 
attentive  reader  may  remark  that  he  practically  abandons  it 
when  he  takes  up  the  subject  of  infinite  series.  De  Morgan's 
theory  is  stated  in  his  volume  on  Trigonometry  and  Double 
Algebra.  In  the  chapter  (of  the  book)  headed  "  On  symbolic 
algebra  "  he  writes:  "  In  abandoning  the  meaning  of  symbols, 
we  also  abandon  those  of  the  words  which  describe  them.  Thus 
addition  is  to  be,  for  the  present,  a  sound  void  of  sense.  It 
is  a  mode  of  combination  represented  by  + ;  when  +  receives 
its  meaning,  so  also  will  the  word  addition.  It  is  most  impor- 
tant that  the  student  should  bear  in  mind  that,  with  one  exception, 
no  word  nor  sign  of  arithmetic  or  algebra  has  one  atom  of  mean- 
ing throughout  this  chapter,  the  object  of  which  is  symbols, 
and  their  laws  of  combination,  giving  a  symbolic  algebra  which 
may  hereafter  become  the  grammar  of  a  hundred  distinct  sig- 
nificant algebras.  If  any  one  were  to  assert  that  -f  and  — 


26  TEN   BRITISH   MATHEMATICIANS 

might  mean  reward  and  punishment,  and  A,  B,  C,  etc.,  might 
stand  for  virtues  and  vices,  the  reader  might  believe  him,  or 
contradict  him,  as  he  pleases,  but  not  out  of  this  chapter.  The 
one  exception  above  noted,  which  has  some  share  of  meaning, 
is  the  sign  =  placed  between  two  symbols  as  in  A  =B.  It  indi- 
cates that  the  two  symbols  have  the  same  resulting  meaning, 
by  whatever  steps  attained.  That  A  and  B,  if  quantities, 
are  the  same  amount  of  quantity;  that  if  operations,  they  are 
of  the  same  effect,  etc." 

Here,  it  may  be  asked,  why  does  the  symbol  =  prove  refrac- 
tory to  the  symbolic  theory?  De  Morgan  admits  that  there 
is  one  exception;  but  an  exception  proves  the  rule,  not  in  the 
usual  but  illogical  sense  of  establishing  it,  but  in  the  old  and 
logical  sense  of  testing  its  validity.  If  an  exception  can  be 
established,  the  rule  must  fall,  or  at  least  must  be  modified. 
Here  I  am  talking  not  of  grammatical  rules,  but  of  the  rules 
of  science  or  nature. 

De  Morgan  proceeds  to  give  an  inventory  of  the  fundamental 
symbols  of  algebra,  and  also  an  inventory  of  the  laws  of  algebra. 
The  symbols  are  o,  i,  +,  —  ,  X,  -T-,  ()°,  and  letters;  these 
only,  all  others  are  derived.  His  inventory  of  the  fundamental 
laws  is  expressed  under  fourteen  heads,  but  some  of  them  are 
merely  definitions.  The  laws  proper  may  be  reduced  to  the 
following,  which,  as  he  admits,  are  not  all  independent  of  one 
another: 


I.  Law  of  signs.     +  +  =  +,  4  —  =  -,  -  +  =  -,  --  =  +,   XX  =  X, 

X  -S~  =  ~  »  ~  X  =  ~  >  ~    ~  =  X- 
II.  Commutative  law.    a+b=b+a,  ab=ba. 

III.  Distributive  law.    a(b+c)=ab+ac. 

IV.  Index  laws.    o*Xac=o6+c,  (o6)c=o6c,  (ab)c=acbc.  ^ 
V.  0—0=0,  0-7-0=1. 

The  last  two  may  be  called  the  rules  of  reduction.  De  Morgan 
professes  to  give  a  complete  inventory  of  the  laws  which  the 
symbols  of  algebra  must  obey,  for  he  says,  "  Any  system  of 
symbols  which  obeys  these  laws  and  no  others,  except  they  be 
formed  by  combination  of  these  laws,  and  which  uses  the  pre- 
ceding symbols  and  no  others,  except  they  be  new  symbols 


AUGUSTUS  DE  MORGAN  27 

invented  in  abbreviation  of  combinations  of  these  symbols,  is 
symbolic  algebra."  From  his  point  of  view,  none  of  the  above 
principles  are  rules;  they  are  formal  laws,  that  is,  arbitrarily 
chosen  relations  to  which  the  algebraic  symbols  must  be  subject. 
He  does  not  mention  the  law,  which  had  already  been  pointed 
out  by  Gregory,  namely,  (a+fy+c  =  a+(b+c),  (ab)c  =  a(bc)  and 
to  which  was  afterwards  given  the  name  of  the  law  of  association. 
If  the  commutative  law  fails,  the  associative  may  hold  good; 
but  not  vice  versa.  It  is  an  unfortunate  thing  for  the  symbolist 
or  formalist  that  in  universal  arithmetic  mn  is  not  equal  to  «m; 
for  then  the  commutative  law  would  have  full  scope.  Why 
does  he  not  give  it  full  scope?  Because  the  foundations  of 
algebra  are,  after  all,  real  not  formal,  material  not  symbolic. 
To  the  formalists  the  index  operations  are  exceedingly  refrac- 
tory, in  consequence  of  which  some  take  no  account  of  them, 
but  relegate  them  to  applied  mathematics.  To  give  an  inventory 
of  the  laws  which  the  symbols  of  algebra  must  obey  is  an  impos- 
sible task,  and  reminds  one  not  a  little  of  the  task  of  those 
philosophers  who  attempt  to  give  an  inventory  of  the  a  priori 
knowledge  of  the  mind. 

De  Morgan's  work  entitled  Trigonometry  and  Double  Algebra 
consists  of  two  parts;  the  former  of  which  is  a  treatise  on 
Trigonometry,  and  the  latter  a  treatise  on  generalized  algebra 
which  he  calls  Double  Algebra.  But  what  is  meant  by  Double 
as  applied  to  algebra?  and  why  should  Trigonometry  be  also 
treated  in  the  same  textbook?  The  first  stage  in  the  develop- 
ment of  algebra  is  arithmetic,  where  numbers  only  appear  and 
symbols  of  operations  such  as  +,  X,  etc.  The  next  stage  is 
universal  arithmetic,  where  letters  appear  instead  of  numbers, 
so  as  to  denote  numbers  universally,  and  the  processes  are  con- 
ducted without  knowing  the  values  of  the  symbols.  Let  a  and 
b  denote  any  numbers;  then  such  an  expression  as  a— b  may 
be  impossible;  so  that  in  universal  arithmetic  there  is  always 
a  proviso,  provided  the  operation  is  possible.  The  third  stage  is 
single  algebra,  where  the  symbol  may  denote  a  quantity  forwards 
or  a  quantity  backwards,  and  is  adequately  represented  by 
segments  on  a  straight  line  passing  through  an  origin.  Negative 


28 


TEN   BRITISH   MATHEMATICIANS 


quantities  are  then  no  longer  impossible;  they  are  represented 
by  the  backward  segment.  But  an  impossibility  still  remains 
in  the  latter  part  of  such  an  expression  as  a+b\/  —  i  which  arises 
in  the  solution  of  the  quadratic  equation.  The  fourth  stage 
is  double  algebra;  the  algebraic  symbol  denotes  in  general  a 
segment  of  a  line  in  a  given  plane;  it  is  a  double  symbol  because 
it  involves  two  specifications,  namely,  length  and  direction; 
and  V  — i  is  interpreted  as  denoting  a  quadrant.  The  expres- 
sion a+&V  — i  then  represents  a  line  in  the  plane  having  an 
abscissa  a  and  an  ordinate  b.  Argand  and  Warren  carried 
double  algebra  so  far;  but  they  were  unable  to  interpret  on 
this  theory  such  an  expression  as  e^^-i.  De  Morgan  attempted 
it  by  reducing  such  an  expression  to  the  form  b+q'V  —  i,  and 
he  considered  that  he  had  shown  that  it  could  be  always  so 
reduced.  The  remarkable  fact  is  that  this  double  algebra 
satisfies  all  the  fundamental  laws  above  enumerated,  and  as 
every  apparently  impossible  combination  of  symbols  has  been 
interpreted  it  looks  like  the  complete  form  of  algebra. 

If  the  above  theory  is  true,  the  next  stage  of  development 
ought  to  be  triple  algebra  and  if  a+b'V  —  i  truly  represents  a  line 
in  a  given  plane,  it  ought  to  be  possible  to  find  a  third  term  which 

added  to  the  above  would  represent  a  line  in_space. Argand 

and  some  others  guessed  that  it  was  #+&V  —  i+cV  —  iV  —  i; 
although  this  contradicts  the  truth  established  by  Euler  that 
\/ —  -iv-l  =  e~**.  De  Morgan  and  many  others  worked  hard 
at  the  problem,  but  nothing  came  of  it  uutil  the  problem  was 
taken  up  by  Hamilton.  We  now  see  the  reason  clearly:  the 
symbol  of  double  algebra  denotes  not  a  length  and  a  direction; 
but  a  multiplier  and  an  angle.  In  it  the  angles  are  confined 
to  one  plane;  hence  the  next  stage  will  be  a  quadruple  algebra, 
when  the  axis  of  the  plane  is  made  variable.  And  this  gives 
the  answer  to  the  first  question;  double  algebra  is  nothing  but 
analytical  plane  trigonometry,  and  this  is  the  reason  why  it 
has  been  found  to  be  the  natural  analysis  for  alternating 
currents.  But  De  Morgan  never  got  this  far;  he  died 
with  the  belief  "  that  double  algebra  must  remain  as  the  full 
development  of  the  conceptions  of  arithmetic,  so  far  as 


AUGUSTUS   DE   MORGAN  29 

those  symbols  are  concerned  which  arithmetic  immediately 
suggests." 

When  the  study  of  mathematics  revived  at  the  University 
of  Cambridge,  so  also  did  the  study  of  logic.  The  moving 
spirit  was  Whewell,  the  Master  of  Trinity  College,  whose  prin- 
cipal writings  were  a  History  of  the  Inductive  Sciences,  and 
Philosophy  of  the  Inductive  Sciences.  Doubtless  De  Morgan 
was  influenced  in  his  logical  investigations  by  Whewell;  but 
other  contemporaries  of  influence  were  Sir  W.  Hamilton  of 
Edinburgh,  and  Professor  Boole  of  Cork.  De  Morgan's  work 
on  Formal  Logic,  published  in  1847,  is  principally  remarkable 
for  his  development  of  the  numerically  definite  syllogism.  The 
followers  of  Aristotle  say  and  say  truly  that  from  two  par- 
ticular propositions  such  as  Some  M' s  are  A's,  and  Some  M's 
are  B's  nothing  follows  of  necessity  about  the  relation  of  the 
A's  and  B's.  But  they  go  further  and  say  in  order  that  any 
relation  about  the  A's  and  B's  may  follow  of  necessity,  the  mid- 
dle term  must  be  taken  universally  in  one  of  the  premises.  De 
Morgan  pointed  out  that  from  Most  M's  are  A's  and  Most  M's 
are  B's  it  follows  of  necessity  that  some  A's  are  B's  and  he 
formulated  the  numerically  definite  syllogism  which  puts  this 
principle  in  exact  quantitative  form.  Suppose  that  the  number 
of  the  M's  is  m,  of  the  M's  that  are  A's  is  a,  and  of  the  M's  that 
are  B's  is  6;  then  there  are  at  least  (a+b  — m)  A's  that  are  .B's. 
Suppose  that  the  number  of  souls  on  board  a  steamer  was  1000, 
that  500  were  in  the  saloon,  and  700  were  lost;  it  follows  of 
necessity,  that  at  least  700+500  —  1000,  that  is,  200,  saloon 
passengers  were  lost.  This  single  principle  suffices  to  prove 
the  validity  of  all  the  Aristotelian  moods;  it  is  therefore  a  funda- 
mental principle  in  necessary  reasoning. 

Here  then  De  Morgan  had  made  a  great  advance  by  intro- 
ducing quantification  of  the  terms.  At  that  time  Sir  W.  Hamilton 
was  teaching  at  Edinburgh  a  doctrine  of  the  quantification  of 
the  predicate,  and  a  correspondence  sprang  up.  However,  De 
Morgan  soon  perceived  that  Hamilton's  quantification  was  of 
a  different  character;  that  it  meant  for  example,  substituting 
the  two  forms  The  whole  of  A  is  the  whole  of  B,  and  The  whole  of 


30  TEN  BRITISH  MATHEMATICIANS 

A  is  a  part  of  B  for  the  Aristotelian  form  All  A's  are  JB's. 
Philosophers  generally  have  a  large  share  of  intolerance;  they 
are  too  apt  to  think  that  they  have  got  hold  of  the  whole  truth, 
and  that  everything  outside  of  their  system  is  error.  Hamilton 
thought  that  he  had  placed  the  keystone  in  the  Aristotelian 
arch,  as  he  phrased  it;  although  it  must  have  been  a  curious 
arch  which  could  stand  2000  years  without  a  keystone.  As 
a  consequence  he  had  no  room  for  De  Morgan's  innovations. 
He  accused  De  Morgan  of  plagiarism,  and  the  controversy 
raged  for  years  in  the  columns  of  the  Athenceum,  and  in  the 
publications  of  the  two  writers. 

The  memoirs  on  logic  which  De  Morgan  contributed  to  the 
Transactions  of  the  Cambridge  Philosophical  Society  subsequent 
to  the  publication  of  his  book  on  Formal  Logic  are  by  far  the 
most  important  contributions  which  he  made  to  the  science, 
especially  his  fourth  memoir,  in  which  he  begins  work  in  the 
broad  field  of  the  logic  of  relatives.  This  is  the  true  field  for  the 
logician  of  the  twentieth  century,  in  which  work  of  the  greatest 
importance  is  to  be  done  towards  improving  language  and 
facilitating  thinking  processes  which  occur  all  the  time  in  prac- 
tical life.  Identity  and  difference  are  the  two  relations  which 
have  been  considered  by  the  logician;  but  there  are  many 
others  equally  deserving  of  study,  such  as  equality,  equivalence, 
consanguinity,  affinity,  etc. 

In  the  introduction  to  the  Budget  of  Paradoxes  De  Morgan 
explains  what  he  means  by  the  word.  "  A  great  many  indi- 
viduals, ever  since  the  rise  of  the  mathematical  method,  have, 
each  for  himself,  attacked  its  direct  and  indirect  consequences. 
I  shall  call  each  of  these  persons  a  paradoxer,  and  his  system  a 
paradox.  I  use  the  word  in  the  old  sense:  a  paradox  is  some- 
thing which  is  apart  from  general  opinion,  either  in  subject 
matter,  method,  or  conclusion.  Many  of  the  things  brought 
forward  would  now  be  called  crotchets,  which  is  the  nearest  word 
we  have  to  old  paradox.  But  there  is  this  difference,  that  by 
calling  a  thing  a  crotchet  we  mean  to  speak  lightly  of  it;  which 
was  not  the  necessary  sense  of  paradox.  Thus  in  the  i6th 
century  many  spoke  of  the  earth's  motion  as  the  paradox  of 


AUGUSTUS  DE  MORGAN  31 

Copernicus  and  held  the  ingenuity  of  that  theory  in  very  high 
esteem,  and  some  I  think  who  even  inclined  towards  it.  In  the 
seventeenth  century  the  depravation  of  meaning  took  place, 
in  England  at  least." 

How  can  the  sound  paradoxer  be  distinguished  from  the  false 
paradoxer?  De  Morgan  supplies  the  following  test :  "  The 
manner  in  which  a  paradoxer  will  show  himself,  as  to  sense 
or  nonsense,  will  not  depend  upon  what  he  maintains,  but  upon 
whether  he  has  or  has  not  made  a  sufficient  knowledge  of  what 
has  been  done  by  others,  especially  as  to  the  mode  of  doing  it, 
a  preliminary  to  inventing  knowledge  for  himself.  .  .  .  New 
knowledge,  when  to  any  purpose,  must  come  by  contemplation 
of  old  knowledge,  in  every  matter  which  concerns  thought; 
mechanical  contrivance  sometimes,  not  very  often,  escapes  this 
rule.  All  the  men  who  are  now  called  discoverers,  in  every 
matter  ruled  by  thought,  have  been  men  versed  in  the  minds 
of  their  predecessors  and  learned  in  what  had  been  before  them. 
There  is  not  one  exception." 

I  remember  that  just  before  the  American  Association  met  at 
Indianapolis  in  1890,  the  local  newspapers  heralded  a  great  discov- 
ery which  was  to  be  laid  before  the  assembled  savants — a  young 
man  living  somewhere  in  the  country  had  squared  the  circle. 
While  the  meeting  was  in  progress  I  observed  a  young  man  going 
about  with  a  roll  of  paper  in  his  hand.  He  spoke  to  me  and  com- 
plained that  the  paper  containing  his  discovery  had  not  been 
received.  I  asked  him  whether  his  object  in  presenting  the 
paper  was  not  to  get  it  read,  printed  and  published  so  that 
everyone  might  inform  himself  of  the  result;  to  all  of  which 
he  assented  readily.  But,  said  I,  many  men  have  worked  at 
this  question,  and  their  results  have  been  tested  fully,  and 
they  are  printed  for  the  benefit  of  anyone  who  can  read;  have 
you  informed  yourself  of  their  results?  To  this  there  was  no 
assent,  but  the  sickly  smile  of  the  false  paradoxer. 

The  Budget  consists  of  a  review  of  a  large  collection  of 
paradoxical  books  which  De  Morgan  had  accumulated  in  his 
own  library,  partly  by  purchase  at  bookstands,  partly  from 
books  sent  to  him  for  review,  partly  from  books  sent  to  him  by 


32  TEN   BRITISH  MATHEMATICIANS 

the  authors.  He  gives  the  following  classification:  squarers  of 
the  circle,  trisectors  of  the  angle,  duplicators  of  the  cube,  con- 
structors of  perpetual  motion,  subverters  of  gravitation,  stag- 
nators  of  the  earth,  builders  of  the  universe.  You  will  still 
find  specimens  of  all  these  classes  in  the  New  World  and  in  the 
new  century. 

De  Morgan  gives  his  personal  knowledge  of  paradoxers.  "  I 
suspect  that  I  know  more  of  the  English  class  than  any  man  in 
Britain.  I  never  kept  any  reckoning:  but  I  know  that  one 
year  with  another — and  less  of  late  years  than  in  earlier  time — 
I  have  talked  to  more  than  five  in  each  year,  giving  more  than 
a  hundred  and  fifty  specimens.  Of  this  I  am  sure,  that  it  is 
my  own  fault  if  they  have  not  been  a  thousand.  Nobody 
knows  how  they  swarm,  except  those  to  whom  they  naturally 
resort.  They  are  in  all  ranks  and  occupations,  of  all  ages  and 
characters.  They  are  very  earnest  people,  and  their  purpose 
is  bona  fide,  the  dissemination  of  their  paradoxes.  A  great  many 
— the  mass,  indeed — are  illiterate,  and  a  great  many  waste  their 
means,  and  are  in  or  approaching  penury.  These  discoverers 
despise  one  another." 

A  paradoxer  to  whom  De  Morgan  paid  the  compliment 
which  Achilles  paid  Hector — to  drag  him  round  the  walls  again 
and  again — was  James  Smith,  a  successful  merchant  of  Liver- 
pool. He  found  ^  =  3^.  His  mode  of  reasoning  was  a  curious 
caricature  of  the  reductio  ad  absurdum  of  Euclid.  He  said  let 
T=3^,  and  then  showed  that  on  that  supposition,  every  other 
value  of  r  must  be  absurd ;  consequently  TT  =  3!  is  the  true  value. 
The  following  is  a  specimen  of  De  Morgan's  dragging  round 
the  walls  of  Troy:  "  Mr.  Smith  continues  to  write  me  long 
letters,  to  which  he  hints  that  I  am  to  answer.  In  his  last  of 
31  closely  written  sides  of  note  paper,  he  informs  me,  with  refer- 
ence to  my  obstinate  silence,  that  though  I  think  myself  and 
am  thought  by  others  to  be  a  mathematical  Goliath,  I  have 
resolved  to  play  the  mathematical  snail,  and  keep  within  my 
shell.  A  mathematical  snail!  This  cannot  be  the  thing  so  called 
which  regulates  the  striking  of  a  clock;  for  it  would  mean  that 
I  am  to  make  Mr.  Smith  sound  the  true  time  of  day,  which  I 


AUGUSTUS  DE   MORGAN  33 

would  by  no  means  undertake  upon  a  clock  that  gains  19  seconds 
odd  in  every  hour  by  false  quadrative  value  of  TT.  But  he 
ventures  to  tell  me  that  pebbles  from  the  sling  of  simple  truth 
and  common  sense  will  ultimately  crack  my  shell,  and  put 
me  hors  de  combat.  The  confusion  of  images  is  amusing: 
Goliath  turning  himself  into  a  snail  to  avoid  ir  =  3|  and  James 
Smith,  Esq.,  of  the  Mersey  Dock  Board:  and  put  hors  de  combat 
by  pebbles  from  a  sling.  If  Goliath  had  crept  into  a  snail  shell, 
David  would  have  cracked  the  Philistine  with  his  foot.  There 
is  something  like  modesty  in  the  implication  that  the  crack-shell 
pebble  has  not  yet  taken  effect;  it  might  have  been  thought 
that  the  slinger  would  by  this  time  have  been  singing — And 
thrice  [and  one-eighth]  I  routed  all  my  foes,  And  thrice  [and  one- 
eighth]  I  slew  the  slain." 

In  the  region  of  pure  mathematics  De  Morgan  could  detect 
easily  the  false  from  the  true  paradox;  but  he  was  not  so  pro- 
ficient in  the  field  of  physics.  His  father-in-law  was  a  para- 
doxer,  and  his  wife  a  paradoxer;  and  in  the  opinion  of  the 
physical  philosophers  De  Morgan  himself  scarcely  escaped. 
His  wife  wrote  a  book  describing  the  phenomena  of  spiritualism, 
table-rapping,  table-turning,  etc.;  and  De  Morgan  wrote  a 
preface  in  which  he  said  that  he  knew  some  of  the  asserted 
facts,  believed  others  on  testimony,  but  did  not  pretend  to  know 
whether  they  were  caused  by  spirits,  or  had  some  unknown 
and  unimagined  origin.  From  this  alternative  he  left  out  ordi- 
nary material  causes.  Faraday  delivered  a  lecture  on  Spirit- 
ualism, in  which  he  laid  it  down  that  in  the  investigation  we 
ought  to  set  out  with  the  idea  of  what  is  physically  possible, 
or  impossible;  De  Morgan  could  not  understand  this. 


SIR  WILLIAM  ROWAN  HAMILTON* 

(1805-1865) 

WILLIAM  ROWAN  HAMILTON  was  born  in  Dublin,  Ireland, 
on  the  3d  of  August,  1805.  His  father,  Archibald  Hamilton, 
was  a  solicitor  in  the  city  of  Dublin;  his  mother,  Sarah  Hutton, 
belonged  to  an  intellectual  family,  but  she  did  not  live  to  exer- 
cise much  influence  on  the  education  of  her  son.  There  has 
been  some  dispute  as  to  how  far  Ireland  can  claim  Hamilton; 
Professor  Tait  of  Edinburgh  in  the  Encyclopaedia  Brittanica 
claims  him  as  a  Scotsman,  while  his  biographer,  the  Rev.  Charles 
Graves,  claims  him  as  essentially  Irish.  The  facts  appear  to 
be  as  follows:  His  father's  mother  was  a  Scotch  woman;  his 
father's  father  was  a  citizen  of  Dublin.  But  the  name  "Hamil- 
ton" points  to  Scottish  origin,  and  Hamilton  himself  said  that  his 
family  claimed  to  have  come  over  from  Scotland  in  the  time 
of  James  I.  Hamilton  always  considered  himself  an  Irishman; 
and  as  Burns  very  early  had  an  ambition  to  achieve  something 
for  the  renown  of  Scotland,  so  Hamilton  in  his  early  years  had 
a  powerful  ambition  to  do  something  for  the  renown  of  Ireland. 
In  later  life  he  used  to  say  that  at  the  beginning  of  the  century 
people  read  French  mathematics,  but  that  at  the  end  of  it  they 
would  be  reading  Irish  mathematics. 

Hamilton,  when  three  years  of  age,  was  placed  in  the  charge 
of  his  uncle,  the  Rev.  James  Hamilton,  who  was  the  curate  of 
Trim,  a  country  town,  about  twenty  miles  from  Dublin,  and 
who  was  also  the  master  of  the  Church  of  England  school. 
From  his  uncle  he  received  all  his  primary  and  secondary  edu- 
cation and  also  instruction  in  Oriental  languages.  As  a  child 
Hamilton  was  a  prodigy;  at  three  years  of  age  he  was  a  superior 
reader  of  English  and  considerably  advanced  in  arithmetic; 

*  This  Lecture  was  delivered  April  16,  1901. — EDITORS. 
34 


SIR   WILLIAM  ROWAN  HAMILTON  35 

at  four  a  good  geographer;  at  five  able  to  read  and  translate 
Latin,  Greek,  and  Hebrew,  and  liked  to  recite  Dryden,  Collins, 
Milton  and  Homer;  at  eight  a  reader  of  Italian  and  French 
and  giving  vent  to  his  feelings  in  extemporized  Latin;  at  ten 
a  student  of  Arabic  and  Sanscrit.  When  twelve  years  old  he 
met  Zerah  Colburn,  the  American  calculating  boy,  and  engaged 
with  him  in  trials  of  arithmetical  skill,  in  which  trials  Hamilton 
came  off  with  honor,  although  Colburn  was  generally  the  victor. 
These  encounters  gave  Hamilton  a  decided  taste  for  arithmetical 
computation,  and  for  many  years  afterwards  he  loved  to  perform 
long  operations  in  arithmetic  in  his  mind,  extracting  the  square 
and  cube  root,  and  solving  problems  that  related  to  the  proper- 
ties of  numbers.  When  thirteen  he  received  his  initiation  into 
algebra  from  Clairault's  Algebra  in  the  French,  and  he  made 
an  epitome,  which  he  ambitiously  entitled  "  A  Compendious 
Treatise  on  Algebra  by  William  Hamilton." 

When  Hamilton  was  fourteen  years  old,  his  father  died  and 
left  his  children  slenderly  provided  for.  Henceforth,  as  the  elder 
brother  of  three  sisters,  Hamilton  had  to  act  as  a  man.  This 
year  he  addressed  a  letter  of  welcome,  written  in  the  Persian 
language,  to  the  Persian  Ambassador  then  on  a  visit  to  Dublin; 
and  he  met  again  Zerah  Colburn.  In  the  interval  Zerah  had 
attended  one  of  the  great  public  schools  of  England.  Hamilton 
had  been  at  a  country  school  in  Ireland,  and  was  now  able  to 
make  a  successful  investigation  of  the  methods  by  which  Zerah 
made  his  lightning  calculations.  When  sixteen,  Hamilton 
studied  the  Differential  Calculus  by  the  help  of  a  French  text- 
book, and  began  the  study  of  the  Mecanique  celeste  of  Laplace, 
and  he  was  able  at  the  beginning  of  this  study  to  detect  a  flaw 
in  the  reasoning  by  which  Laplace  demonstrates  the  theorem  of 
the  parallelogram  of  forces.  This  criticism  brought  him  to  the 
notice  of  Dr.  Brinkley,  who  was  then  the  professor  of  astronomy 
in  the  University  of  Dublin,  and  resided  at  Dunkirk,  about 
five  miles  from  the  centre  of  the  city.  He  also  began  an  inves- 
tigation for  himself  of  equations  which  represent  systems  of 
straight  lines  in  a  plane,  and  in  so  doing  hit  upon  ideas  which 
he  afterwards  developed  into  his  first  mathematical  memoir  to 


36  TEN   BRITISH   MATHEMATICIANS 

the  Royal  Irish  Academy.  Dr.  Brinkley  is  said  to  have 
remarked  of  him  at  this  time:  "This  young  man,  I  do  not 
say  will  be,  but  is,  the  first  mathematician  of  his  age." 

At  the  age  of  eighteen  Hamilton  entered  Trinity  College, 
Dublin,  the  University  of  Dublin  founded  by  Queen  Elizabeth, 
and  differing  from  the  Universities  of  Oxford  and  Cambridge 
in  having  only  one  college.  Unlike  Oxford,  which  has  always 
given  prominence  to  classics,  and  Cambridge,  which  has  always 
given  prominence  to  mathematics,  Dublin  at  that  time  gave 
equal  prominence  to  classics  and  to  mathematics.  In  his  first 
year  Hamilton  won  the  very  rare  honor  of  optime  at  his  exami- 
nation in  Homer.  In  the  old  Universities  marks  used  to  be  and 
in  some  cases  still  are  published,  descending  not  in  percentages 
but  by  means  of  the  scale  of  Latin  adjectives:  optime,  valdebene, 
bene,  satis,  mediocriter,  mx  medi,  non;  optime  means  passed 
with  the  very  highest  distinction;  mx  means  passed  but  with 
great  difficulty.  This  scale  is  still  in  use  in  the  medical  exami- 
nations of  the  University  of  Edinburgh.  Before  entering  col- 
lege Hamilton  had  been  accustomed  to  translate  Homer  into 
blank  verse,  comparing  his  result  with  the  translations  of  Pope 
and  Cowper;  and  he  had  already  produced  some  original 
poems.  In  this,  his  first  year  he  wrote  a  poem  "  On  college 
ambition  "  which  is  a  fair  specimen  of  his  poetical  attainments. 

Oh!  Ambition  hath  its  hour 

Of  deep  and  spirit-stirring  power; 

Not  in  the  tented  field  alone, 

Nor  peer-engirded  court  and  throne; 

Nor  the  intrigues  of  busy  life ; 

But  ardent  Boyhood's  generous  strife, 

While  yet  the  Enthusiast  spirit  turns 

Where'er  the  light  of  Glory  burns, 

Thinks  not  how  transient  is  the  blaze, 

But  longs  to  barter  Life  for  Praise. 

Look  round  the  arena,  and  ye  spy 
Pallid  cheek  and  faded  eye; 
Among  the  bands  of  rivals,  few 
Keep  their  native  healthy  hue: 
Night  and  thought  have  stolen  away 


SIR  WILLIAM   ROWAN  HAMILTON  37 

Their  once  elastic  spirit's  play. 
A  few  short  hours  and  all  is  o'er, 
Some  shall  win  one  triumph  more; 
Some  from  the  place  of  contest  go 
Again  defeated,  sad  and  slow. 

What  shall  reward  the  conqueror  then 

For  all  his  toil,  for  all  his  pain, 

For  every  midnight  throb  that  stole 

So  often  o'er  his  fevered  soul? 

Is  it  the  applaudings  loud 

Or  wond'ring  gazes  of  the  crowd; 

Disappointed  envy's  shame, 

Or  hollow  voice  of  fickle  Fame? 

These  may  extort  the  sudden  smile, 

May  swell  the  heart  a  little  while; 

But  they  leave  no  joy  behind, 

Breathe  no  pure  transport  o'er  the  mind, 

Nor  will  the  thought  of  selfish  gladness 

Expand  the  brow  of  secret  sadness. 

Yet  if  Ambition  hath  its  hour 
Of  deep  and  spirit-stirring  power, 
Some  bright  rewards  are  all  its  own, 
And  bless  its  votaries  alone: 
The  anxious  friend's  approving  eye; 
The  generous  rivals'  sympathy; 
And  that  best  and  sweetest  prize 
Given  by  silent  Beauty's  eyes ! 
These  are  transports  true  and  strong, 
Deeply  felt,  remembered  long: 
Time  and  sorrow  passing  o'er 
Endear  their  memory  but  the  more. 

The  "silent  Beauty"  was  not  an  abstraction,  but  a  young 
lady  whose  brothers  were  fellow-students  of  Trinity  College. 
This  led  to  much  effusion  of  poetry;  but  unfortunately  while 
Hamilton  was  writing  poetry  about  her  another  young  man 
was  talking  prose  to  her;  with  the  result  that  Hamilton  experi- 
enced a  disappointment.  On  account  of  his  self-consciousness, 
inseparable  probably  from  his  genius,  he  felt  the  disappointment 
keenly.  He  was  then  known  to  the  professor  of  astronomy, 
and  walking  from  the  College  to  the  Observatory  along  the  Royal 


38  TEN  BRITISH  MATHEMATICIANS 

Canal,  he  was  actually  tempted  to  terminate  his  life  in  the 
water. 

In  his  second  year  he  formed  the  plan  of  reading  so  as 
to  compete  for  the  highest  honors  both  in  classics  and  in 
mathematics.  At  graduation  two  gold  medals  were  awarded, 
the  one  for  distinction  in  classics,  the  other  for  distinction  in 
mathematics.  Hamilton  aimed  at  carrying  off  both.  In  his 
junior  year  he  received  an  optime  in  mathematical  physics; 
and,  as  the  winner_of  two  optimes,  the  one  in  classics,  the  other 
in  mathematics,  he  immediately  became  a  celebrity  in  the  intel- 
lectual circle  of  Dublin. 

In  his  senior  year  he  presented  to  the  Royal  Irish  Academy 
a  memoir  embodying  his  research  on  systems  of  lines.  He  now 
called  it  a  "Theory  of  Systems  of  Rays  "  and  it  was  printed 
in  the  Transactions.  About  this  time  Dr.  Brinkley  was  ap- 
pointed to  the  bishopric  of  Cloyne,  and  in  consequence  resigned 
the  professorship  of  astronomy.  In  the  United  Kingdom  it  is 
customary  when  a  post  becomes  vacant  for  aspirants  to  lodge 
a  formal  application  with  the  appointing  board  and  to  sup- 
plement their  own  application  by  testimonial  letters  from  com- 
petent authorities.  In  the  present  case  quite  a  number  of  can- 
didates appeared,  among  them  Airy,  who  afterwards  became 
Astronomer  Royal  of  England,  and  several  Fellows  of  Trinity 
College,  Dublin.  Hamilton  did  not  become  a  formal  candidate, 
but  he  was  invited  to  apply,  with  the  result  that  he  received 
the  appointment  while  still  an  undergraduate,  and  not  twenty- 
two  years  of  age.  Thus  was  his  undergraduate  career  signalized 
much  more  than  by  the  carrying  off  of  the  two  gold  medals. 
Before  assuming  the  duties  of  his  chair  he  made  a  tour  through 
England  and  Scotland,  and  met  for  the  first  time  the  poet 
Wordsworth  at  his  home  at  Rydal  Mount,  in  Cumberland. 
They  had  a  midnight  walk,  oscillating  backwards  and  forwards 
between  Rydal  and  Ambleside,  absorbed  in  converse  on  high 
themes,  and  finding  it  almost  impossible  to  part.  Wordsworth 
afterwards  said  that  Coleridge  and  Hamilton  were  the  two 
most  wonderful  men,  taking  all  their  endowments  together, 
that  he  had  ever  met. 


' 

SIR  WILLIAM  ROWAN  HAMILTON  39 

In  October,  1827,  he  came  to  reside  at  the  place  which  was 
destined  to  be  the  scene  of  his  scientific  labors.  I  had  the 
pleasure  of  visiting  it  last  summer  as  the  guest  of  his  successor. 
The  Observatory  is  situated  on  the  top  of  a  hill,  Dunsink,  about 
five  miles  from  Dublin.  The  house  adjoins  the  observatory; 
to  the  east  is  an  extensive  lawn;  to  the  west  a  garden  with  stone 
wall  and  shaded  walks;  to  the  south  a  terraced  field;  at  the  foot 
of  the  hill  is  the  Royal  Canal;  to  the  southeast  the  city  of 
Dublin;  while  the  view  is  bounded  by  the  sea  and  the  Dublin 
and  Wicklow  Mountains ;  a  fine  home  for  a  poet  or  a  philosopher 
or  a  mathematician,  and  in  Hamilton  all  three  were  combined. 

Settled  at  the  Observatory  he  started  out  diligently  as  an 
observer,  but  he  found  it  difficult  to  stand  the  low  temperatures 
incident  to  the  work.  He  never  attained  skill  as  an  observer, 
and  unfortunately  he  depended  on  a  very  poor  assistant.  Him- 
self a  brilliant  computer,  with  a  good  observer  for  assistant, 
the  work  of  the  observatory  ought  to  have  flourished.  One  of 
the  first  distinguished  visitors  at  the  Observatory  was  the  poet 
Wordsworth,  in  commemoration  of  which  one  of  the  shaded 
walks  in  the  garden  was  named  Wordsworth's  walk.  Words- 
worth advised  him  to  concentrate  his  powers  on  science;  and, 
not  long  after,  wrote  him  as  follows:  "You  send  me  showers 
of  verses  which  I  receive  with  much  pleasure,  as  do  we  all :  yet 
have  we  fears  that  this  employment  may  seduce  you  from  the 
path  of  science  which  you  seem  destined  to  tread  with  so  much 
honor  to  yourself  and  profit  to  others.  Again  and  again  I  must 
repeat  that  the  composition  of  verse  is  infinitely  more  of  an 
art  than  men  are  prepared  to  believe,  and  absolute  success  in 
it  depends  upon  innumerable  minutiae  which  it  grieves  me  you 
should  stoop  to  acquire  a  knowledge  of.  .  .  Again  I  do  venture 
to  submit  to  your  consideration,  whether  the  poetical  parts  of 
your  nature  would  not  find  a  field  more  _favorable  to  their 
exercise  in  the  regions  of  prose;  not  because  those  regions  are 
humbler,  but  because  they  may  be  gracefully  and  profitably 
trod,  with  footsteps  less  careful  and  in  measures  less  elaborate." 

Hamilton  possessed  the  poetic  imagination;  what  he  was 
deficient  in  was  the  technique  of  the  poet.  The  imagination 


40  TEN   BRITISH   MATHEMATICIANS 

of  the  poet  is  kin  to  the  imagination  of  the  mathematician; 
both  extract  the  ideal  from  a  mass  of  circumstances.  In  this 
connection  De  Morgan  wrote:  "  The  moving  power  of  mathe- 
tical  invention  is  not  reasoning  but  imagination.  We  no  longer 
apply  the  homely  term  maker  in  literal  translation  of  poet]  but 
discoverers  of  all  kinds,  whatever  may  be  their  lines,  are  makers, 
or,  as  we  now  say,  have  the  creative  genius."  Hamilton  spoke 
of  the  Mecanique  analytique  of  Lagrange  as  a  "scientific  poem"; 
Hamilton  himself  was  styled  the  Irish  Lagrange.  Engineers 
venerate  Rankine,  electricians  venerate  Maxwell;  both  were 
scientific  discoverers  and  likewise  poets,  that^is,  amateur  poets. 
The  proximate  cause  of  the  shower  of  verses  was  that  Hamilton 
had  fallen  hi  love  for  the  second  time.  The  young  lady  was 
Miss  de  Vere,  daughter  of  an  accomplished  Irish  baronet,  and 
who  like  Tennyson's  Lady  Clara  Vere  de  Vere  could  look  back 
on  a  long  and  illustrious  descent.  Hamilton  had  a  pupil  in 
Lord  Adare,  the  eldest  son  of  the  Earl  of  Dunraven,  and  it  was 
while  visiting  Adare  Manor  that  he  was  introduced  to  the  De 
Vere  family,  who  lived  near  by  at  Curragh  Chase.  His  suit 
was  encouraged  by  the  Countess  of  Dunraven,  it  was  favorably 
received  by  both  father  and  mother,  he  had  written  many  sonnets 
of  which  Ellen  de  Vere  was  the  inspiration,  he  had  discussed 
with  her  astronomy,  poetry  and  philosophy;  and  was  on  the 
eve  of  proposing  when  he  gave  up  because  the  young  lady  inci- 
dentally said  to  him  that  "she  could  not  live  happily  anywhere 
but  at  Curragh."  His  action  shows  the  working  of  a  too  self- 
conscious  mind,  proud  of  his  own  intellectual  achievements, 
and  too  much  awed  by  her  long  descent.  So  he  failed  for  the 
second  time;  but  both  of  these  ladies  were  friends  of  his  to  the 
last. 

At  the  age  of  27  he  contributed  to  the  Irish  Academy  a 
supplementary  paper  on  his  Theory  of  Systems  of  Rays,  in  which 
he  predicted  the  phenomenon  of  conical  refraction;  namely, 
that,  under  certain  conditions  a  single  ray  incident  on  a  biaxial 
crystal  would  be  broken  up  into  a  cone  of  rays,  and  likewise 
that  under  certain  conditions  a  single  emergent  ray  would 
appear  as  a  cone  of  rays.  The  prediction  was  made  by  Hamilton 


SIR  WILLIAM   ROWAN   HAMILTON  41 

on  Oct.  22nd;  it  was  experimentally  verified  by  his  colleague 
Prof.  Lloyd  on  Dec.  i4th.  It  is  not  experiment  alone  or 
mathematical  reasoning  alone  which  has  built  up  the  splendid 
temple  of  physical  science,  but  the  two  working  together;  and 
of  this  we  have  a  notable  exemplification  in  the  discovery  of 
conical  refraction. 

Twice  Hamilton  chose  well  but  failed ;  now  he  made  another 
choice  and  succeeded.  The  lady  was  a  Miss  Bayly,  who 
visited  at  the  home  of  her  sister  near  Dunsink  hill.  The  lady 
had  serious  misgivings  about  the  state  of  her  health;  but  the 
marriage  took  place.  The  kind  of  wife  which  Hamilton  needed 
was  one  who  could  govern  him  and  efficiently  supervise  all 
domestic  matters;  but  the  wife  he  chose  was,  from  weakness 
of  body  and  mind,  incapable  of  doing  it.  As  a  consequence, 
Hamilton  worked  for  the  rest  of  his  life  under  domestic  dif- 
ficulties of  no  ordinary  kind. 

At  the  age  of  28  he  made  a  notable  addition  to  the  theory 
of  Dynamics  by  extending  to  it  the  idea  of  a  Characteristic 
Function,  which  he  had  previously  applied  with  success  to 
the  science  of  Optics  in  his  Theory  of  Systems  of  Rays.  It 
was  contributed  to  the  Royal  Society  of  London,  and  printed 
in  their  Philosophical  Transactions.  The  Royal  Society  of 
London  is  the  great  scientific  society  of  England,  founded 
in  the  reign  of  Charles  II,  and  of  which  Newton  was  one  of 
the  early  presidents;  Hamilton  was  invited  to  become  a  fellow 
but  did  not  accept,  as  he  could  not  afford  the  expense. 

At  the  age  of  29  he  read  a  paper  before  the  Royal  Irish 
Academy,  which  set  forth  the  result  of  long  meditation  and 
investigation  on  the  nature  of  Algebra  as  a  science;  the  paper 
is  entitled  "  Algebra  as  the  Science  of  Pure  Time."  The  main 
idea  is  that  as  Geometry  considered  as  a  science  is  founded 
upon  the  pure  intuition  of  space,  so  algebra  as  a  science  is 
founded  upon  the  pure  intuition  of  time.  He  was  never  satis- 
fied with  Peacock's  theory  of  algebra  as  a  "  System  of  Signs 
and  their  Combinations  ";  nor  with  De  Morgan's  improve- 
ment of  it;  he  demanded  a  more  real  foundation.  In  reading 
Kant's  Critique  of  Pure  Reason  he  was  struck  by  the  follow- 


42  TEN   BRITISH   MATHEMATICIANS 

ing  passage:  "  Time  and  space  are  two  sources  of  knowledge 
from  which  various  a  priori  synthetical  cognitions  can  be 
derived.  Of  this,  pure  mathematics  gives  a  splendid  example 
in  the  case  of  our  cognitions  of  space  and  its  various  relations. 
As  they  are  both  pure  forms  of  sensuous  intuition,  they  render 
synthetical  propositions  a  priori  possible."  Thus,  according 
to  Kant,  space  and  time  are  forms  of  the  intellect;  and  Ham- 
ilton reasoned  that,  as  geometry  is  the  science  of  the  former, 
so  algebra  must  be  the  science  of  the  latter.  When  algebra 
is  based  on  any  unidimensional  subject,  such  as  time,  or  a 
straight  line,  a  difficulty  arises  in  explaining  the  roots  of  a 
quadratic  equation  when  they  are  imaginary.  To  get  over 
this  difficulty  Hamilton  invented  a  theory  of  algebraic  couplets, 
which  has  proved  a  conundrum  in  the  mathematical  world. 
Some  20  years  ago  there  flourished  in  Edinburgh  a  mathematician 
named  Sang  who  had  computed  the  most  elaborate  tables 
of  logarithms  in  existence— which  still  exist  in  manuscript. 
On  reading  the  theory  in  question  he  first  judged  that  either 
HamDton  was  crazy,  or  else  that  he  (Sang)  was  crazy,  but 
eventually  reached  the  more  comforting  alternative.  On  the 
other  hand,  Prof.  Tait  believes  in  its  soundness,  and  endeavors 
to  bring  it  down  to  the  ordinary  comprehension. 

We  have  seen  that  the  British  Association  for  the  Advance- 
ment of  Science  was  founded  in  1831,  and  that  its  first  meeting 
was  in  the  ancient  city  of  York.  It  was  a  policy  of  the  founders 
not  to  meet  in  London,  but  in  the  provincial  cities,  so  that 
thereby  greater  interest  in  the  advance  of  science  might  be 
produced  over  the  whole  land.  The  cities  chosen  for  the  place 
of  meeting  in  following  years  were  the  University  towns:  Ox- 
ford, Cambridge,  Edinburgh,  Dublin.  Hamilton  was  the  only 
representative  of  Ireland  present  at  the  Oxford  meeting;  and 
at  the  Oxford,  Cambridge,  and  Edinburgh  meetings  he  not 
only  contributed  scientific  papers,  but  he  acquired  renown 
as  a  scientific  orator.  In  the  case  of  the  Dublin  meeting 
he  was  chief  organizer  beforehand,  and  chief  orator  when 
it  met.  The  week  of  science  was  closed  by  a  grand 
dinner  given  in  the  library  of  Trinity  College;  and  an 


SIR  WILLIAM  ROWAN  HAMILTON  43 

incident  took  place  which  is  thus  described  by  an  American 
scientist : 

"  We  assembled  in  the  imposing  hall  of  Trinity  Library, 
two  hundred  and  eighty  feet   long,  at  six  o'clock.     When  the 
company  was   principally  assembled,  I  observed  a   little  stir 
near  the  place  where   I  stood,  which    nobody  could  explain, 
and  which,  in  fact,  was  not  comprehended  by  more  than  two 
or  three  persons  present.     In  a  moment,  however,  I  perceived 
myself  standing  near  the  Lord  Lieutenant  and  his  suite,  in 
front  of  whom  a  space  had  been  cleared,  and  by  whom  was 
Professor    Hamilton,  looking    very    much    embarrassed.     The 
Lord  Lieutenant  then  called  him  by  name,   and  he  stepped 
into  the  vacant  space.     '  I  am,'  said  his  Excellency,  '  about 
to  exercise  a  prerogative  of  royalty,  and  it  gives  me  great 
pleasure  to  do  it,  on  this  splendid  public  occasion,  which  has 
brought  together  so  many  distinguished  men  from  all  parts 
of  the  empire,  and  from  all  parts  even  of  the  world  where 
science  is  held  in  honor.     But,  in  exercising  it,  Professor  Ham- 
ilton, I  do  not  confer  a  distinction.     I  but  set  the  royal,  and 
therefore  the  national  mark  on  a  distinction  already  acquired 
by  your  genius  and  labors.'     He  went   on   in  this  way  for 
three  of  four  minutes,  his  voice  very  fine,  rich  and  full;    his 
manner  as  graceful  and  dignified  as  possible;   and  his  language 
and  allusions  appropriate  and  combined  into  very  ample  flow- 
ing sentences.    Then,  receiving  the  State  sword  from  one  of 
his  attendants,  he  said,  '  Kneel  down,  Professor  Hamilton  '; 
and  laying  the  blade  gracefully  and  gently  first  on  one  shoulder, 
and  then  on  the  other,  he  said,  '  Rise  up,  Sir  William  Rowan 
Hamilton.'    The  Knight  rose,  and  the  Lord  Lieutenant  then 
went  up,  and  with  an  appearance  of  great  tact  in  his  manner, 
shook   hands   with   him.     No   reply   was   made.    The   whole 
scene  was  imposing,  rendered  so,  partly  by  the  ceremony  itself, 
but  more  by  the  place  in  which  it  passed,  by  the  body  of  very 
distinguished  men  who  were  assembled  there,  and  especially 
by  the  extraordinarily  dignified  and  beautiful  manner  in  which 
it  was  performed  by  the  Lord  Lieutenant.     The  effect  at  the 
time  was  great,  and  the  general  impression  was  that,  as  the 


44  TEN  BRITISH   MATHEMATICIANS 

ft 

honor  was  certainly  merited  by  him  who  received  it,  so  the 
words  by  which  it  was  conferred  were  so  graceful  and  appro- 
priate that  they  constituted  a  distinction  by  themselves,  greater 
than  the  distinction  of  knighthood.  I  was  afterwards  told 
that  this  was  the  first  instance  in  which  a  person  had  been 
knighted  by  a  Lord  Lieutenant  either  for  scientific  or  literary 
merit." 

Two  years  after  another  great  honor  came  to  Hamilton 
— the  presidency  of  the  Royal  Irish  Academy.  While  holding 
this  office,  in  the  year  1843,  when  38  years  old,  he  made  the 
discovery  which  will  ever  be  considered  his  highest  title  to 
fame.  The  story  of  the  discovery  is  told  by  Hamilton  him- 
self in  a  letter  to  his  son:  "  On  the  i6th  day  of  October,  which 
happened  to  be  a  Monday,  and  Council  day  of  the  Royal 
Irish  Academy,  I  was  walking  in  to  attend  and  preside,  and 
your  mother  was  walking  with  me  along  the  Royal  Canal, 
to  which  she  had  perhaps  driven;  and  although  she  talked  with 
me  now  and  then,  yet  an  undercurrent  of  thought  was  going 
on  in  my  mind,  which  gave  at  last  a  result,  whereof  it  is  not 
too  much  to  say  that  I  felt  at  once  the  importance.  An  electric 
circuit  seemed  to  close;  and  a  spark  flashed  forth,  the  herald 
(as  I  foresaw  immediately)  of  many  long  years  to  come  of 
definitely  directed  thought  and  work,  by  myself  if  spared,  and 
at  all  events  on  the  part  of  others,  if  I  should  even  be  allowed 
to  live  long  enough  distinctly  to  communicate  the  discovery. 
Nor  could  I  resist  the  impulse — unphilosophical  as  it  may 
have  been — to  cut  with  a  knife  on  a  stone  of  Brougham  Bridge, 
as  we  passed  it,  the  fundamental  formula  with  the  symbols 
*, ;',  k;  namely, 

•O«OTO**7 

1?=j2=-k2  —  Jjk=.  _T> 

which  contains  the  solution  of  the  problem,  but  of  course  as 
an  inscription  has  long  since  mouldered  away.  A  more  durable 
notice  remains,  however,  in  the  Council  Book  of  the  Academy 
for  that  day,  which  records  the  fact  that  I  then  asked  for  and 
obtained  leave  to  read  a  paper  on  Quaternions,  at  the  first 
general  meeting  of  the  session,  which  reading  took  place  accord- 
ingly on  Monday  the  i3th  of  November  following." 


SIR  WILLIAM   ROWAN  HAMILTON  45 

Last  summer  Prof.  Joly  and  I  took  the  walk  here  described. 
We  started  from  the  Observatory,  walked  down  the  terraced 
field,  then  along  the  path  by  the  side  of  the  Royal  Canal 
towards  Dublin  until  we  came  to  the  second  bridge  spanning 
the  canal.  The  path  of  course  goes  under  the  Bridge,  and 
the  inner  side  of  the  Bridge  presents  a  very  convenient  surface 
for  an  inscription.  I  have  seen  this  incident  quoted  as  an 
example  of  how  a  genius  strikes  on  a  discovery  all  of  a  sudden. 
No  doubt  a  problem  was  solved  then  and  there,  but  the  problem 
had  engaged  Hamilton's  thoughts  and  researches  for  fifteen 
years.  It  is  rather  an  illustration  of  how  genius  is  patience, 
or  a  faculty  for  infinite  labor.  What  was  Hamilton  struggling 
to  do  all  these  years?  To  emerge  from  Flatland  into  Space; 
in  other  words,  Algebra  had  been  extended  so  as  to  apply  to 
lines  in  a  plane;  but  no  one  had  been  able  to  extend  it  so  as  to 
apply  to  lines  in  space.  The  greatness-  of  the  feat  is  made 
evident  by  the  fact  that  most  analysts  are  still  crawling  in 
Flatland.  The  same  year  in  which  he  discovered  Quaternions 
the  Government  granted  him  a  pension  of  £200  per  annum 
for  life,  on  account  of  his  scientific  work. 

We  have  seen  how  Hamilton  gained  two  oplimes,  one  in  clas- 
sics, the  other  in  physics,  the  highest  possible  distinction  in  his 
college  course;  how  he  was  appointed  professor  of  astronomy 
while  yet  an  undergraduate;  how  he  was  a  scientific  chief  in  the 
British  Association  at  27;  how  he  was  knighted  for  his  scientific 
achievements  at  30;  how  he  was  appointed  president  of  the 
Royal  Irish  Academy  at  32;  how  he  discovered  Quaternions 
and  received  a  Government  pension  at  38;  can  you  imagine 
that  this  brilliant  and  successful  genius  would  fall  a  victim  to 
intemperance?  About  this  time  at  a  dinner  of  a  scientific  society 
in  Dublin  he  lost  control  of  himself,  and  was  so  mortified  that, 
on  the  advice  of  friends  he  resolved  to  abstain  totally.  This 
resolution  he  kept  for  two  years;  when  happening  to  be  a 
member  of  a  scientific  party  at  the  castle  of  Lord  Rosse,  an 
amateur  astronomer  then  the  possessor  of  the  largest  telescope 
in  existence,  he  was  taunted  for  sticking  to  water,  particularly 
by  Airy  the  Greenwich  astronomer.  He  broke  his  good  reso- 


46  TEN   BRITISH   MATHEMATICIANS 

lution,  and  from  that  time  forward  the  craving  for  alcoholic 
stimulants  clung  to  him.  How  could  Hamilton  with  all  his 
noble  aspirations  fall  into  such  a  vice?  The  explanation  lay 
in  the  want  of  order  which  reigned  in  his  home.  He  had  no 
regular  times  for  his  meals;  frequently  had  no  regular  meals 
at  all,  but  resorted  to  the  sideboard  when  hunger  compelled 
him.  What  more  natural  in  such  condition  than  that  he  should 
refresh  himself  with  a  quaff  of  that  beverage  for  which  Dublin 
is  famous — porter  labelled  X3?  After  Hamilton's  death  the 
dining-room  was  found  covered  with  huge  piles  of  manuscript, 
with  convenient  walks  between  the  piles;  when  these  literary 
remains  were  wheeled  out  and  examined,  china  plates  with  the 
relics  of  food  upon  them  were  found  between  the  sheets  of 
manuscript,  plates  sufficient  in  number  to  furnish  a  kitchen. 
He  used  to  carry  on,  says  his  eldest  son,  long  trains  of  algebraical 
and  arithmetical  calculations  in  his  mind,  during  which  he  was 
unconscious  of  the  earthly  necessity  of  eating;  "we  used  to  bring 
in  a  '  snack  '  and  leave  it  in  his  study,  but  a  brief  nod  of 
recognition  of  the  intrusion  of  the  chop  or  cutlet  was  often  the 
only  result,  and  his  thoughts  went  on  soaring  upwards." 

In  1845  Hamilton  attended  the  second  Cambridge  meeting 
of  the  British  Association;  and  after  the  meeting  he  was  lodged 
for  a  week  in  the  rooms  in  Trinity  College  which  tradition 
points  out  as  those  in  which  Sir  Isaac  Newton  composed  the 
Principia.  This  incident  was  intended  as  a  compliment  and  it 
seems  to  have  impressed  Hamilton  powerfully.  He  came  back 
to  the  Observatory  with  the  fixed  purpose  of  preparing  a  work 
on  Quaternions  which  might  not  unworthily  compare  with  the 
Principia  of  Newton,  and  in  order  to  obtain  more  leisure  for  this 
undertaking  he  resigned  the  office  of  president  of  the  Royal 
Irish  Academy.  He  first  of  all  set  himself  to  the  preparation 
of  a  course  of  lectures  on  Quaternions,  which  were  delivered  in 
Trinity  College,  Dublin,  in  1848,  and  were  six  in  number.  Among 
his  hearers  were  George  Salmon,  now  well  known  for  his  highly 
successful  series  of  manuals  on  Analytical  Geometry;  and  Arthur 
Cayley,  then  a  Fellow  of  Trinity  College,  Cambridge.  These 
lectures  were  afterward  expanded  and  published  in  1853,  under 


SIR   WILLIAM  ROWAN  HAMILTON  47 

the  title  of  Lectures  on  Quaternions,  at  the  expense  of  Trinity 
College,  Dublin.  Hamilton  had  never  had  much  experience 
as  a  teacher;  the  volume  was  criticised  for  diffuseness  of  style, 
and  certainly  Hamilton  sometimes  forgot  the  expositor  in  the 
orator.  The  book  was  a  paradox — a  sound  paradox,  and  of 
his  experience  as  a  paradoxer  Hamilton  wrote:  "It  required  a 
certain  capital  of  scientific  reputation,  amassed  in  former  years, 
to  make  it  other  than  dangerously  imprudent  to  hazard  the 
publication  of  a  work  which  has,  although  at  bottom  quite 
conservative,  a  highly  revolutionary  air.  It  was  part  of  the 
ordeal  through  which  I  had  to  pass,  an  episode  in  the  battle 
of  life,  to  know  that  even  candid  and  friendly  people  secretly 
or,  as  it  might  happen,  openly,  censured  or 
ridiculed  me,  for  what  appeared  to  them 
my  monstrous  innovations."  One  of  these 
monstrous  innovations  was  the  principle  that 
ij  is  not  =ji  but  =  —ji;  the  truth  of  which 
is  evident  from  the  diagram.  Critics  said 
that  he  held  that  3X4  is  not  =4X3;  which  proceeds  on  the 
assumption  that  only  numbers  can  be  represented  by  letter 
symbols. 

Soon  after  the  publication  of  the  Lectures,  he  became  aware 
of  itfe  imperfection  as  a  manual  of  instruction,  and  he  set  him- 
self to  prepare  a  second  book  on  the  model  of  Euclid's  Elements. 
He  estimated  that  it  would  fill  400  pages  and  take  two  years 
to  prepare;  it  amounted  to  nearly  800  closely  printed  pages 
and  took  seven  years.  At  times  he  would  work  for  twelve 
hours  on  a  stretch;  and  he  also  suffered  from  anxiety  as  to  the 
means  of  publication.  Trinity  College  advanced  £200,  he  paid 
£50  out  of  his  own  pocket,  but  when  illness  came  upon  him  the 
expense  of  paper  and  printing  had  mounted  up  to  £400.  He 
was  seized  by  an  acute  attack  of  gout,  from  which,  after 
several  months  of  suffering,  he  died  on  Sept.  2,  1865,  in  the 
6ist  year  of  his  age. 

,  It  is  pleasant  to  know  that  this  great  mathematician  received 
during  his  last  illness  an  honor  from  the  United  States,  which 
made  him  feel  that  he  had  realized  the  aim  of  his  great  labors. 


48  TEN   BRITISH  MATHEMATICIANS 

While  the  war  between  the  North  and  South  was  in  progress, 
the  National  Academy  of  Sciences  was  founded,  and  the  news 
which  came  to  Hamilton  was  that  he  had  been  elected  one  of 
ten  foreign  members,  and  that  his  name  had  been  voted  to 
occupy  the  specially  honorable  position  of  first  on  the  list.  Sir 
William  Rowan  Hamilton  was  thus  the  first  foreign  associate 
of  the  National  Academy  of  Sciences  of  the  United  States. 

As  regards  religion  Hamilton  was  deeply  reverential  in 
nature.  He  was  born  and  brought  up  in  the  Church  of  England, 
which  was  then  the  established  Church  in  Ireland.  He  lived 
in  the  time  of  the  Oxford  movement,  and  for  some  time  he 
sympathized  with  it;  but  when  several  of  his  friends,  among 
them  the  brother  of  Miss  De  Vere,  passed  over  into  the  Roman 
Catholic  Church,  he  modified  his  opinion  of  the  movement  and 
remained  Protestant  to  the  end. 

The  immense  intellectual  activity  of  Hamilton,  especially  dur- 
ing the  years  when  he  was  engaged  on  the  enormous  labor  of 
writing  the  Elements  of  Quaternions,  made  him  a  recluse,  and 
necessarily  took  away  from  his  power  of  attending  to  the  prac- 
tical affairs  of  life.  Some  said  that  however  great  a  master  of 
pure  time  he  might  be  he  was  not  a  master  of  sublunary  time. 
His  neighbors  also  took  advantage  of  his  goodness  of  heart. 
Surrounding  the  house  there  is  an  extensive  lawn  affording  good 
pasture,  and  on  it  Hamilton  pastured  a  cow.  A  neighbor  advised 
Hamilton  that  his  cow  would  be  much  better  contented  by 
having  another  cow  for  company  and  bargained  with  Hamilton 
to  furnish  the  companion  provided  Hamilton  paid  something 
like  a  dollar  per  month. 

Here  is  Hamilton's  own  estimate  of  himself.  "  I  have 
very  long  admired  Ptolemy's  description  of  his  great  astronomical 
master,  Hipparchus,  as  avyp  <£iXo7iwos  K<U  ^iXaAi^fys;  a  labor- 
loving  and  truth-loving  man.  Be  such  my  epitaph." 

Hamilton's  family  consisted  of  two  sons  and  one  daughter. 
At  the  time  of  his  death,  the  Elements  of  Quaternions  was 
all  finished  excepting  one  chapter.  His  eldest  son,  William 
Edwin  Hamilton,  wrote  a  preface,  and  the  volume  was  pub- 
lished at  the  expense  of  Trinity  College,  Dublin.  Only  500 


SIR   WILLIAM  ROWAN  HAMILTON  49 

copies  were  printed,  and  many  of  those  were  presented.  In 
consequence  it  soon  became  a  scarce  book,  and  as  much  as 
$35.00  has  been  paid  for  a  copy.  A  new  edition,  in  two  volumes, 
is  now  being  published  by  Prof.  Joly,  his  successor  in  Dunsink 
Observatory. 


GEORGE  BOOLE* 

(1815-1864) 

GEORGE  BOOLE  was  born  at  Lincoln,  England,  on  the  2d 
of  November,  1815.  His  father,  a  tradesman  of  very  limited 
means,  was  attached  to  the  pursuit  of  science,  particularly 
of  mathematics,  and  was  skilled  in  the  construction  of  optical 
instruments.  Boole  received  his  elementary  education  at  the 
National  School  of  the  city,  and  afterwards  at  a  commercial 
school;  but  it  was  his  father  who  instructed  him  in  the  elements 
of  mathematics,  and  also  gave  him  a  taste  for  the  construction 
and  adaptation  of  optical  instruments.  However,  his  early 
ambition  did  not  urge  him  to  the  further  prosecution  of  mathe- 
mathical  studies,  but  rather  to  becoming  proficient  in  the 
ancient  classical  languages.  In  this  direction  he  could  receive  no 
help  from  his  father,  but  to  a  friendly  bookseller  of  the  neigh- 
borhood he  was  indebted  for  instruction  in  the  rudiments  of 
the  Latin  Grammar.  To  the  study  of  Latin  he  soon  added 
that  of  Greek  without  any  external  assistance;  and  for  some 
years  he  perused  every  Greek  or  Latin  author  that  came  within 
his  reach.  At  the  early  age  of  twelve  his  proficiency  in  Latin 
made  him  the  occasion  of  a  literary  controversy  in  his  native 
city.  He  produced  a  metrical  translation  of  an  ode  of  Horace, 
which  his  father  in  the  pride  of  his  heart  inserted  in  a  local 
journal,  stating  the  age  of  the  translator.  A  neighboring 
school-master  wrote  a  letter  to  the  journal  in  which  he  denied, 
from  internal  evidence,  that  the  version  could  have  been  the 
work  of  one  so  young.  In  his  early  thirst  for  knowledge  of 
languages  and  ambition  to  excel  in  verse  he  was  like  Hamilton, 
but  poor  Boole  was  much  more  heavily  oppressed  by  the  res 
angusta  domi — the  hard  conditions  of  his  home.  Accident 

*  This  Lecture  was  delivered  April  19,  1901. — EDITORS. 


GEORGE   BOOLE  51 

discovered  to  him  certain  defects  in  his  methods  of  classical 
study,  inseparable  from  the  want  of  proper  early  training,  and 
it  cost  him  two  years  of  incessant  labor  to  correct  them. 

Between  the  ages  of  sixteen  and  twenty  he  taught  school 
as  an  assistant  teacher,  first  at  Doncaster  in  Yorkshire,  after- 
wards at  Waddington  near  Lincoln;  and  the  leisure  of  these 
years  he  devoted  mainly  to  the  study  of  the  principal  modern 
languages,  and  of  patristic  literature  with  the  view  of  studying 
to  take  orders  in  the  Church.  This  design,  however,  was  not 
carried  out,  owing  to  the  financial  circumstances  of  his  parents 
and  some  other  difficulties.  In  his  twentieth  year  he  de- 
cided on  opening  a  school  on  his  own  account  in  his  native 
city;  thenceforth  he  devoted  all  the  leisure  he  could  com- 
mand to  the  study  of  the  higher  mathematics,  and  solely  with 
the  aid  of  such  books  as  he  could  procure.  Without  other 
assistance  or  guide  he  worked  his  way  onward,  and  it  was  his 
own  opinion  that  he  had  lost  five  years  of  educational  progress 
by  his  imperfect  methods  of  study,  and  the  want  of  a  helping 
hand  to  get  him  over  difficulties.  No  doubt  it  cost  him  much 
time;  but  when  he  had  finished  studying  he  was  already  not 
only  learned  but  an  experienced  investigator. 

We  have  seen  that  at  this  time  (1835)  the  great  masters 
of  mathematical  analysis  wrote  in  the  French  language;  and 
Boole  was  naturally  led  to  the  study  of  the  Mecanique  celeste 
of  Laplace,  and  the  Mecanique  analytique  of  Lagrange.  While 
studying  the  latter  work  he  made  notes  from  which  there 
eventually  emerged  his  first  mathematical  memoir,  entitled, 
"  On  certain  theorems  in  the  calculus  of  variations."  By  the 
same  works  his  attention  was  attracted  to  the  transformation 
of  homogeneous  functions  by  linear  substitu  ions,  and  in  the 
course  of  his  subsequent  investigations  he  was  led  to  results 
which  are  now  regarded  as  the  foundation  of  the  modern 
Higher  Algebra.  In  the  publication  of  his  results  he  received 
friendly  assistance  from  D.  F.  Gregory,  a  younger  member  of 
the  Cambridge  school,  and  editor  of  the  newly  founded  Cam- 
bridge Mathematical  Journal.  Gregory  and  other  friends  sug- 
gested that  Boole  should  take  the  regular  mathematical  course 


52  TEN   BRITISH   MATHEMATICIANS 

at  Cambridge,  but  this  he  was  unable  to  do;  he  continued  to 
teach  school  for  his  own  support  and  that  of  his  aged  parents, 
and  to  cultivate  mathematical  analysis  in  the  leisure  left  by 
a  laborious  occupation. 

Duncan  F.  Gregory  was  one  of  a  Scottish  family  already 
distinguished  in  the  annals  of  science.  His  grandfather  was 
James  Gregory,  the  inventor  of  the  refracting  telescope  and 
discoverer  of  a  convergent  series  for  TT.  A  cousin  of  his  father 
was  David  Gregory,  a  special  friend  and  fellow  worker  of  Sir 
Isaac  Newton.  D.  F.  Gregory  graduated  at  Cambridge,  and 
after  graduation  he  immediately  turned  his  attention  to  the 
logical  foundations  of  analysis.  He  had  before  him  Peacock's 
theory  of  algebra,  and  he  knew  that  in  the  analysis  as  devel- 
oped by  the  French  school  there  were  many  remarkable  phe- 
nomena awaiting  explana  on;  particularly  theorems  which 
involved  what  was  cal  ed  the  separation  of  symbols.  He 
embodied  his  results  in  a  paper  "  On  the  real  Nature  of  sym- 
bolical Algebra  "  which  was  printed  in  the  Transactions  of 
the  Royal  Society  of  Edinburgh. 

Boole  became  a  master  of  the  method  of  separation  of 
symbols,  and  by  attempting  to  apply  it  to  the  solution  of 
differential  equations  with  variable  coefficients  was  led  to  devise 
a  general  method  in  analysis.  The  account  of  it  was  printed 
in  the  Transactions  of  the  Royal  Society  of  London,  and  brought 
its  author  a  Royal  medal.  Boole's  study  of  the  separation 
of  symbols  naturally  led  him  to  a  study  of  the  foundations 
of  analysis,  and  he  had  before  him  the  writings  of  Peacock, 
Gregory  and  De  Morgan.  He  was  led  to  entertain  very  wide 
views  of  the  domain  of  mathematical  analysis;  in  fact  that  it 
was  coextensive  with  exact  analysis,  and  so  embraced  formal 
logic.  In  1848,  as  we  have  seen,  the  controversy  arose  be- 
tween Hamilton  and  De  Morgan  about  the  quantification  of 
terms;  the  general  interest  which  that  controversy  awoke  in 
the  relation  of  mathematics  to  logic  induced  Boole  to  prepare 
for  publication  his  views  on  the  subject,  which  he  did  that 
same  year  in  a  small  volume  entitled  Mathematical  Analysis 
of  Logic. 


GEORGE  BOOLE  53 

About  this  time  what  are  denominated  the  Queen's  Colleges 
of  Ireland  were  instituted  at  Belfast,  Cork  and  Galway;  and 
in  1849  Boole  was  appointed  to  the  chair  of  mathematics  in 
the  Queen's  College  at  Cork.  In  this  more  suitable  environ- 
ment he  set  himself  to  the  preparation  of  a  more  elaborate 
work  on  the  mathematical  analysis  of  logic.  For  this  pur- 
pose he  read  extensively  books  on  psychology  and  logic,  and 
as  a  result  published  in  1854  the  work  on  which  .his  fame 
chiefly  rests — "  An  Investigation  of  the  Laws  of  Thought,  on 
which  are  founded  the  mathematical  theories  of  logic  and 
probabilities."  Subsequently  he  prepared  textbooks  on  Dif- 
ferential Equations  and  Finite  Differences;  the  former  of  which 
remained  the  best  English  textbook  on  its  subject  until  the 
publication  of  Forsyth's  Differential  Equations. 

Prefixed  to  the  Laws  of  Thought  is  a  dedication  to  Dr. 
Ryall,  Vice-President  and  Professor  of  Greek  in  the  same 
College.  In  the  following  year,  perhaps  as  a  result  of  the  ded- 
ication, he  married  Miss  Everest,  the  niece  of  that  colleague. 
Honors  came:  Dublin  University  made  him  an  LL.D.,  Oxford 
a  D.C.L.;  and  the  Royal  Society  of  London  elected  him  a 
Fellow.  But  Boole's  career  was  cut  short  in  the  midst  of  his 
usefulness  and  scientific  labors.  One  day  in  1864  he  walked 
from  his  residence  to  the  College,  a  distance  of  two  miles,  in 
a  drenching  rain,  and  lectured  in  wet  clothes.  The  result 
was  a  feverish  cold  which  soon  fell  upon  his  lungs  and  terminated 
his  career  on  December  8,  1864,  in  the  5oth  year  of  his  age. 

De  Morgan  was  the  man  best  qualified  to  judge  of  the  value 
of  Boole's  work  in  the  field  of  logic;  and  he  gave  it  generous 
praise  and  help.  In  writing  to  the  Dublin  Hamilton  he  said, 
"  I  shall  be  glad  to  see  his  work  (Laws  of  Thought)  out,  f°r 
he  has,  I  think,  got  hold  of  the  true  connection  of  algebra  and 
logic."  At  another  time  he  wrote  to  the  same  as  follows: 
"  All  metaphysicians  except  you  and  I  and  Boole  consider 
mathematics  as  four  books  of  Euclid  and  algebra  up  to  quad- 
ratic equations."  We  might  infer  that  these  three  contem- 
porary mathematicians  who  were  likewise  philosophers  would 
form  a  triangle  of  friends.  But  it  was  not  so;  Hamilton  was 


64  TEN  BRITISH  MATHEMATICIANS 

a  friend  of  De  Morgan,  and  De  Morgan  a  friend  of  Boole; 
but  the  relation  of  friend,  although  convertible,  is  not  neces- 
sarily transitive.  Hamilton  met  De  Morgan  only  once  in  his 
life,  Boole  on  the  other  hand  with  comparative  frequency; 
yet  he  had  a  voluminous  correspondence  with  the  former 
extending  over  20  years,  but  almost  no  correspondence  with 
the  latter.  De  Morgan's  investigations  of  double  algebra  and 
triple  algebra  prepared  him  to  appreciate  the  quaternions, 
whereas  Boole  was  too  much  given  over  to  the  symbolic  theory 
to  appreciate  geometric  algebra. 

Hamilton's  biography  has  appeared  in  three  volumes,  pre- 
pared by  his  friend  Rev.  Charles  Graves ;  De  Morgan's  biography 
has  appeared  in  one  volume,  prepared  by  his  widow;  of  Boole 
no  biography  has  appeared.  A  biographical  notice  of  Boole 
was  written  for  the  Proceedings  of  the  Royal  Society  of  London 
by  his  friend  the  Rev.  Robert  Harley,  and  it  is  to  it  that  I 
am  indebted  for  most  of  my  biographical  data.  Last  summer 
when  in  England  I  learned  that  the  reason  why  no  adequate 
biography  of  Boole  had  appeared  was  the  unfortunate  temper 
and  lack  of  sound  judgment  of  his  widow.  Since  her  hus- 
band's death  Mrs.  Boole  has  published  a  paradoxical  book 
of  the  false  kind  worthy  of  a  notice  in  De  Morgan's  Budget. 

The  work  done  by  Boole  in  applying  mathematical  analysis 
to  logic  necessarily  led  him  to  consider  the  general  question 
of  how  reasoning  is  accomplished  by  means  of  symbols.  The 
view  which  he  adopted  on  this  point  is  stated  at  page  68  of  the 
Laws  of  Thought.  "  The  conditions  of  valid  reasoning  by  the 
aid  of  symbols,  are :  First,  that  a  fixed  interpretation  be  assigned 
to  the  symbols  employed  in  the  expression  of  the  data;  and 
that  -the  laws  of  the  combination  of  those  symbols  be  correctly 
determined  from  that  interpretation;  Second,  that  the  formal 
processes  of  solution  or  demonstration  be  conducted  throughout 
in  obedience  to  all  the  laws  determined  as  above,  without 
regard  to  the  question  of  the  interpretability  of  the  particular 
results  obtained;  Third,  that  the  final  result  be  interpretable 
in  form,  and  that  it  be  actually  interpreted  in  accordance 
with  that  system  of  interpretation  which  has  been  employed 


GEORGE  BOOLE  55 

in  the  expression  of  the  data."  As  regards  these  conditions 
it  may  be  observed  that  they  are  very  different  from  the  for- 
malist view  of  Peacock  and  De  Morgan,  and  that  they  incline 
towards  a  realistic  view  of  analysis,  as  held  by  Hamilton.  True 
he  speaks  of  interpretation  instead  of  meaning,  but  it  is  a 
fixed  interpretation;  and  the  rules  for  the  processes  of  solution 
are  not  to  be  chosen  arbitrarily,  but  are  to  be  found  out  from 
the  particular  system  of  interpretation  of  the  symbols. 

It  is  Boole's  second  condition  which  chiefly  calls  for  study 
and  examination;  respecting  it  he  observes  as  follows:  "The 
principle  in  question  may  be  considered  as  resting  upon  a  gen- 
eral law  of  the  mind,  the  knowledge  of  which  is  not  given  to 
us  a  priori,  that  is,  antecedently  to  experience,  but  is  derived, 
like  the  knowledge  of  the  other  laws  of  the  mind,  from  the 
clear  manifestation  of  the  general  principle  in  the  particular 
instance.  A  single  example  of  reasoning,  in  which  symbols 
are  employed  in  obedience  to  laws  founded  upon  their  inter- 
pretation, but  without  any  sustained  reference  to  that  inter- 
pretation, the  chain  of  demonstration  conducting  us  through 
intermediate  steps  which  are  not  interpretable  to  a'  final  result 
which  is  interpretable,  seems  not  only  to  establish  the  validity 
of  the  particular  application,  but  to  make  known  to  us  the 
general  law  manifested  therein.  No  accumulation  of  instances 
can  properly  add  weight  to  such  evidence.  It  may  furnish 
us  with  clearer  conceptions  of  that  common  element  of  truth 
upon  which  the  application  of  the  principle  depends,  and  so 
prepare  the  way  for  its  reception.  It  may,  where  the  imme- 
diate force  of  the  evidence  is  not  felt,  serve  as  a  verification, 
a  posteriori,  of  the  practical  validity  of  the  principle  in  ques- 
tion. But  this  does  not  affect  the  position  affirmed,  viz.,  that 
the  general  principle  must  be  seen  in  the  particular  instance — 
seen  to  be  general  in  application  as  well  as  true  in  the  special 
example.  The  employment  of  the  uninterpretable  symbol  Vi  — 
in  the  intermediate  processes  of  trigonometry  furnishes  an 
illustration  of  what  has  been  said.  I  apprehend  that  there  is 
no  mode  of  explaining  that  application  which  does  not  covertly 
assume  the  very  principle  in  question.  But  that  principle, 


56  TEN  BRITISH  MATHEMATICIANS 

though  not,  as  I  conceive,  warranted  by  formal  reasoning 
based  upon  other  grounds,  seems  to  deserve  a  place  among  those 
axiomatic  truths  which  constitute  in  some  sense  the  foundation 
of  general  knowledge,  and  which  may  properly  be  regarded 
as  expressions  of  the  mind's  own  laws  and  constitution." 

We  are  all  familiar  with  the  fact  that  algebraic  reasoning 
may  be  conducted  through  intermediate  equations  without 
requiring  a  sustained  reference  to  the  meaning  of  these  equa- 
tions; but  it  is  paradoxical  to  say  that  these  equations  can,  in 
any  case,  have  no  meaning  or  interpretation.  It  may  not  be 
necessary  to  consider  their  meaning,  it  may  even  be  difficult 
to  find  their  meaning,  but  that  they  have  a  meaning  is  a  dic- 
tate of  common  sense.  It  is  entirely  paradoxical  to  say  that, 
as  a  general  process,  we  can  start  from  equations  having  a 
meaning,  and  arrive  at  equations  having  a  meaning  by  passing 
through  equations  which  have  no  meaning.  The  particular 
instance  in  which  Boole  sees' the  truth  of  the  paradoxical  prin- 
ciple is  the  successful  employment  of  the  uninterpretable  sym- 
bol V  — i  in  the  intermediate  processes  of  trigonometry.  So 
soon  then  'as  this  symbol  is  interpreted,  or  rather,  so  soon  as 
its  meaning  is  demonstrated,  the  evidence  for  the  principle 
fails,  and  Boole's  transcendental  logic  falls. 

In  the  algebra  of  quantity  we  start  from  elementary  symbols 
denoting  numbers,  but  are  soon  led  to  compound  forms  which 
do  not  reduce  to  numbers;  so  in  the  algebra  of  logic  we  start 
from  elementary  symbols  denoting  classes,  but  are  soon  intro- 
duced to  compound  expressions  which  cannot  be  reduced  to 
simple  classes.  Most  mathematical  logicians  say,  Stop,  we  do 
not  know  what  this  combination  means.  Boole  says,  It  may 
be  meaningless,  go  ahead  all  the  same.  The  design  of  the  Laws 
of  Thought  is  stated  by  the  author  to  be  to  investigate  the 
fundamental  laws  of  those  operations  of  the  mind  by  which 
reasoning  is  performed;  to  give  expression  to  them  in  the  sym- 
bolical language  of  a  Calculus,  and  upon  this  foundation  to 
establish  the  Science  of  Logic  and  construct  its  method;  to 
make  that  method  itself  the  basis  of  a  general  method  for  the 
application  of  the  mathematical  doctrine  of  Probabilities;  and, 


GEORGE  BOOLE  57 

finally  to  collect  from  the  various  elements  of  truth  brought  to 
view  in  the  course  of  these  inquiries  some  probable  intimations 
concerning  the  nature  and  constitution  of  the  human  mind. 

Boole's  inventory  of  the  symbols  required  in  the  algebra  of 
logic  is  as  follows:  first,  Literal  symbols,  as  x,  y,  etc.,  representing 
things  as  subjects  of  our  conceptions;  second,  Signs  of  operation, 
as  +,  — ,  X,  standing  for  those  operations  of  the  mind  by  which 
the  conceptions  of  things  are  combined  or  resolved  so  as  to  form 
new  conceptions  involving  the  same  elements;  third,  The  sign 
of  identity  =  ;  not  equality  merely,  but  identity  which  involves 
equality.  The  symbols  x,  y,  etc.,  are  used  to  denote  classes; 
and  it  is  one  of  Boole's  maxims  that  substantives  and  adjectives 
alike  denote  classes.  "  They  may  be  regarded,"  he  says,  "  as 
differing  only  in  this  respect,  that  the  former  expresses  the  sub- 
stantive existence  of  the  individual  thing  or  things  to  which 
it  refers,  the. latter  implies  that  existence.  If  we  attach  to  the 
adjective  the  universally  understood  subject,  "  being "  or 
"  thing/'  it  becomes  virtually  a  substantive,  and  may  for  all  the 
essential  purposes  of  reasoning  be  replaced  by  the  substantive." 
Let  us  then  agree  to  represent  the  class  of  individuals  to  which  a 
particular  name  is  applicable  by-  a  single  letter  as  x.  If  the 
name  is  men  for  instance,  Jet  x  represent  all  men,  or  the  class 
men.  Again,  if  an  adjective,  as  good,  is  employed  as  a  term  of 
description,  let  us  represent  by  a  letter,  as  y,  all  things  to  which 
the  description  good  is  applicable,  that  is,  all  good  things  or  the 
class  good  things.  Then  the  combination  yx  will  represent 
good  men. 

Boole's  symbolic  logic  was  brought  to  my  notice  by  Pro- 
fessor Tait,  when  I  was  a  student  in  the  physical  laboratory  of 
Edinburgh  University.  I  studied  the  Laws  of  Thought  and  I 
found  that  those  who  had  written  on  it  regarded  the  method 
as  highly  mysterious;  the  results  wonderful,  but  the  processes 
obscure.  I  reduced  everything  to  diagram  and  model,  and  1 
ventured  to  publish  my  views  on  the  subject  in  a  small  volume 
called  Principles  of  the  Algebra  of  Logic;  one  of  the  chief  points 
I  made  is  the  philological  and  analytical  difference  between 
the  substantive  and  the  adjective.  What  I  said  was  that  the 


58  TEN   BRITISH   MATHEMATICIANS 

word  man  denotes  a  class,  but  the  word  white  does  not;  in  the 
former  a  definite  unit-object  is  specified,  in  the  latter  no  unit- 
object  is  specified.  We  can  exhibit  a  type  of  a  man,  we  cannot 
exhibit  a  type  of  a  white. 

The  identification  of  the  substantive  and  adjective  on 
the  one  hand  and  their  discrimination  on  the  other  hand, 
lead  to  different  conceptions  of  what  De  Morgan  called  the 
universe.  Boole's  conception  of  the  Universe  is  as  follows 
(Laws  of  Thought,  p.  42) :  "  In  every  discourse,  whether  of 
the  mind  conversing  with  its  own  thoughts,  or  of  the  indi- 
vidual in  his  intercourse  with  others,  there  is  an  assumed  or 
expressed  limit  within  which  the  subjects  of  its  operation  are 
confined.  The  most  unfettered  discourse  is  that  in  which  the 
words  we  use  are  understood  in  the  widest  possible  application, 
and  for  them  the  limits  of  discourse  are  coextensive  with  those 
of  the  universe  itself.  But  more  usually  we  confine  ourselves 
to  a  less  spacious  field.  Sometimes  in  discoursing  of  men  we 
imply  (without  expressing  the  limitation)  that  it  is  of  men  only 
under  certain  circumstances  and  conditions  that  we  speak,  as 
of  civilized  men,  or  of  men  in  the  vigor  of  life,  or  of  men  under 
some  other  condition  or  relation.  Now,  whatever  may  be  the 
extent  of  the  field  within  which  all  the  objects  of  our  discourse 
are  found,  that  field  may  properly  be  termed  the  universe  of 
discourse." 

Another  view  leads  to  the  conception  of  the  Universe  as  a 
collection  of  homogeneous  units,  which  may  be  finite  or  infinite 
in  number;  and  in  a  particular  problem  the  mind  considers  the 
relation  of  identity  between  different  groups  of  this  collection. 
This  universe  corresponds  to  the  series  of  events,  in  the  theory 
of  Probability;  and  the  characters  correspond  to  the  different 
ways  in  which  the  event  may  happen.  The  difference  is  that 
the  Algebra  of  Logic  considers  necessary  data  and  relations; 
while  the  theory  of  Probability  considers  probable  data  and 
relations.  I  will  explain  the  elements  of  Boole's  method  on  this 
theory. 

The  square  is  a  collection  of  points :  it  may  serve  to  represent 
any  collection  of  homogeneous  units,  whether  finite  or  infinite 


GEORGE   BOOLE 


59 


FIG. i. 


in  number,  that  is,  the  universe  of  the  problem.    Let  x  denote 

inside    the  left-hand  circle,  and  y  inside  the  right-hand  circle. 

Uxy  will  denote  the   points  inside  both  circles 

(Fig.  i).     In  arithmetical  value  x  may  range  from 

i  to  o;  so  also  y,  while  xy  cannot  be  greater  than 

x  or  y,  or  less  than  o  or  x-\-y  —  i.     This  last  is  the 

principle  of  the  syllogism.     From  the  co-ordinate 

nature  of  the  operations  x  and  y,  it  is  evident 

that  Uxy  =  Uyx;  but  this  is  a  different  thing  from  commuting, 

as  Boole  does,  the  relation  of  U  and  x,  which  is  not  that  of 

co-ordination,  but  of  subordination  of  x  to  U,  and  which  is 

properly  denoted  by  writing  U  first. 

Suppose  y  to  be  the  same  character  as  x;  we  will  then  always 
have  Uxx=Ux;  that  is,  an  elementary  selective  symbol  x  is 
always  such  that  x2  =  x.  These  are  but  the  symbols  of  ordinary 
algebra  which  satisfy  this  relation,  namely  i  and  o;  these  are 
also  the  extreme  selective  symbols  all  and  none.  The  law  in 
question  was  considered  Boole's  paradox;  it  plays  a  very  great 
part  in  the  development  of  his  method. 

Let  Uxy  =  Uz,  where  =  means  identical  with, 
not  equal  to;  we  may  write  xy  =  z}  leaving  the  U 
to  be  understood.  It  does  not  mean  that  the 
combination  of  characters  xy  is  identical  with  the 
character  z;  but  that  those  points  which  have  the 
characters  x  and  y  are  identical  with  the  points 
which  have  the  character  z  (Fig.  2).  From  xy  =  z,  we  derive 

x  =  -z;  what  is  the  meaning  of  this  expression?    We  shall  return 

to  the  question,  after  we  have  considered  +  and  — . 

Let  us  now  consider  the  expression  U(x-\-y). 
If  the  x  points  and  the  y  points  are  outside  of  one 
another,  it  means  the  sum  of  the  x  points  and  the 
y  points  (Fig.  3).  So  far  all  are  agreed.  But 
suppose  that  the  x  points  and  the  y  points  are 
partially  identical  (Fig.  4);  then  there  arises 
difference  of  opinion.  Boole  held  that  the  common  points  must 
be  taken  twice  over,  or  in  other  words  that  the  symbols  x  and 


FIG.  2. 


FIG.  3. 


60  TEN   BRITISH   MATHEMATICIANS 

y  must  be  treated  all  the  same  as  if  they  were  independent  of 
one  another;  otherwise,  he  held,  no  general  analysis  is  possible. 
U(x-\-y)  will  not  in  general  denote  a  single  class 
of  points;  it  will  involve  in  general  a  duplica- 
tion. 

Similarly,  Boole  held  that  the  expression 
U(x—y)  does  not  involve  the  condition  of  the 
Uy  Deing  wholly  included  in  the  Ux  (Fig.  5). 
If  that  condition  is  satisfied,  U(x—y)  denotes  a  simple  class; 
namely,  the  Ux's  without  the  Uy's.  But  when  there  is  partial 
coincidence  (as  in  Fig.  4),  the  common  points  will  be  cancelled, 
and  the  result  will  be  the  Ux's  which  are  not  y  taken  posi- 
tively and  the  Uy's  which  are  not  x  taken  negatively.  In 
Boole's  view  U(x—y)  was  in  general  an  intermediate  uninter- 
pretable  form,  which  might  be  used  in  reasoning  the  same  way 
as  analysts  used  V—  i. 

Most  of  the  mathematical  logicians  who  have  come  after 
Boole  are  men  who  would  have  stuck  at  the  impossible  sub- 
traction in  ordinary  algebra.  They  say  virtually,  "  How  can 
you  throw  into  a  heap  the  same  things  twice  over;  and  how 
can  you  take  from  a  heap  things  that  are  not  there."  Their 
great  principle  is  the  impossibility  of  taking  the  pants  from  a 
Highlander.  Their  only  conception  of  the  analytical  processes 
of  addition  and  subtraction  is  throwing  into  a  heap  and  taking 
out  of  a  heap.  It  does  not  occur  to  them  that  the  processes  of 
algebra  are  ideal,  and  not  subject  to  gross  material  restr'ctions. 
If  x+y  denotes  a  quality  without  duplication,  it  will  sat- 
isfy the  condition 


but  x2=x,  y2=y, 

2xy  =  o. 
Similarly,  if  x—  y  denote  a  simple  quality,  then 

(x-y)2=x-y, 
x2-\-y2=2xy=x—y, 


GEORGE  BOOLE  61 

therefore,  y  —  2xy  =  —y, 

.-.     y=xy. 

In  other  words,  the  Uy  must  be  included  in  the  Ux  (Fig.  5). 
Here  we  have  assumed   that  the  law  of  signs  is 
the  same  as  in  ordinary  algebra,  and  the  result 
comes  out  correct. 

Suppose  Uz=Uxy;  then  Ux=U-z.    How  are 

y 

the  Ux's  related  to  the  Uy's  and  the  Uz's?    From          ™  5' 
the  diagram  (in  Fig.  2)  we  see  that  the  Ux's  are  identical  with 
all  the  Uyz's  together  with  an  indefinite  portion  of  the  E/'s,  which 
are  neither  y  nor  z.     Boole  discovered  a  general  method  for 
finding  the    meaning  of   any   function   of  elementary  logical 
symbols,  which  applied  to  the  above  case,  is  as  follows: 
When  y  is  an  elementary  symbol, 

•L=y+(T.-y). 
Similarly  1  =  2+ (1—2). 

/.     i  =yz+y(i  -2)  +  (i  -y)z+(i  -y)(i  —z), 

which  means  that  the  CTs  either  have  both  qualities  y  and  2, 
or  y  but  not  2,  or  z  but  not  y,  or  neither  y  and  2.     Let 

-z  =  Ayz+By(i  —2)  +C(i  —  y)z+D(i  —y)(i  —2), 

y 

it  is  required  to  determine  the  coefficients  A,  B,  C,  D.     Sup- 
pose y  =  i,  2  =  1;   then  i—A.     Suppose  y  =  i,  2=0,   then  o=B. 

Suppose  y  =  o,  2  =  1;    then  -  =  C,  and  C  is  infinite;    therefore 

o 

(i— 37)2  =  0;   which  we  see  to  be  true  from  the  diagram.     Sup- 
pose y  —  o,  2  =  0;  then  -=D,  or  Z)  is  indeterminate.    Hence 
o 

-2  =  3>2+an  indefinite  portion  of  (i—  y}(i—  z). 
****** 

Boole  attached  great  importance  to  the  index  law  x2=x. 
He  held  that  it  expressed  a  law  of  thought,  and  formed  the 


62  TEN   BRITISH   MATHEMATICIANS 

characteristic  distinction  of  the  operations  of  the  mind  in  its 
ordinary  discourse  and  reasoning,  as  compared  with  its  oper- 
ations when  occupied  with  the  general  algebra  of  quantity. 
It  makes  possible,  he  said,  the  solution  of  a  quintic  or  equation 
of  higher  degree,  when  the  symbols  are  logical.  He  deduces  from 
it  the  axiom  of  metaphysicians  which  is  termed  the  principle 
of  contradiction,  and  which  affirms  that  it  is  impossible  for 
any  being  to  possess  a  quality,  and  at  the  same  time  not  to 
possess  it.  Let  x  denote  an  elementary  quality  applicable 
to  the  universe  U;  then  i—  x  denotes  the  absence  of  that 
quality.  But  if  x2  =  x,  then  o  =  x— x2,  o  =  x(i—x},  that  is, 
from  Ux2  =  Ux  we  deduce  Ux(i  —  x)  =o. 

He  considers  x(i—x)=o  as  an  expression  of  the  prin- 
ciple of  contradiction.  He  proceeds  to  remark :  "  The  above 
interpretation  has  been  introduced  not  on  account  of  its  imme- 
diate value  in  the  present  system,  but  as  an  illustration  of  a 
significant  fact  in  the  philosophy  of  the  intellectual  powers, 
viz.,  that  what  has  been  commonly  regarded  as  the  funda- 
mental axiom  of  metaphysics  is  but  the  consequence  of  a  law 
of  thought,  mathematical  in  its  form.  I  desire  to  direct  atten- 
tion also  to  the  circumstance  that  the  equation  in  which  that 
fundamental  law  of  thought  is  expressed  is  an  equation  of 
the  second  degree.  Without  speculating  at  all  in  this  chapter 
upon  the  question  whether  that  circumstance  is  necessary  in 
its  own  nature,  we  may  venture  to  assert  that  if  it  had  not 
existed,  the  whole  procedure  of  the  understanding  would  have 
been  different  from  what  it  is." 

We  have  seen  that  De  Morgan  investigated  long  and 
published  much  on  mathematical  logic.  His  logical  writings 
are  characterized  by  a  display  of  many  symbols,  new  alike 
to  logic  and  to  mathematics;  in  the  words  of  Sir  W.  Hamilton 
of  Edinburgh,  they  are  "  horrent  with  mysterious  spiculfe." 
It  was  the  great  merit  of  Boole's  work  that  he  used  the  immense 
power  of  the  ordinary  algebraic  notation  as  an  exact  language, 
and  proved  its  power  for  making  ordinary  language  more 
exact.  De  Morgan  could  well  appreciate  the  magnitude  of 
the  feat,  and  he  gave  generous  testimony  to  it  as  follows: 


GEORGE   BOOLE  63 

"  Boole's  system  of  logic  is  but  one  of  many  proofs  of  genius 
and  patience  combined.  I  might  legitimately  have  entered 
it  among  my  paradoxes,  or  things  counter  to  general  opinion: 
but  it  is  a  paradox  which,  like  that  of  Copernicus,  excited 
admiration  from  its  first  appearance.  That  the  symbolic 
processes  of  algebra,  invented  as  tools  of  numerical  calcula- 
tion, should  be  competent  to  express  every  act  of  thought, 
and  to  furnish  the  grammar  and  dictionary  of  an  all-containing 
system  of  logic,  would  not  have  been  believed  until  it  was 
proved.  When  Hobbes,  in  the  time  of  the  Commonwealth, 
published  his  "  Computation  or  Logique  "  he  had  a  remote 
glimpse  of  some  of  the  points  which  are  placed  in  the  light  of 
day  by  Mr.  Boole.  The  unity  of  the  forms  of  thought  in  all 
the  applications  of  reason,  however  remotely  separated,  wfl 
one  day  be  matter  of  notoriety  and  common  wonder:  and 
Boole's  name  will  be  remembered  in  connection  with  one  of 
the  most  important  steps  towards  the  attainment  of  this 
knowledge." 


ARTHUR  CAYLEY* 

(1821-1895) 

ARTHUR  CAYLEY  was  born  at  Richmond  in  Surrey,  England, 
on  August  16,  1821.  His  father,  Henry  Cayley,  was  descended 
from  an  ancient  Yorkshire  family,  but  had  settled  in  St.  Peters- 
burg, Russia,  as  a  merchant.  His  mother  was  Maria  Antonia 
Doughty,  a  daughter  of  William  Doughty;  who,  according 
to  some  writers,  was  a  Russian;  but  her  father's  name  indi- 
cates an  English  origin.  Arthur  spent  the  first  eight  years  of 
his  life  in  St.  Petersburg.  In  1829  his  parents  took  up  their 
permanent  abode  at  Blackheath,  near  London;  and  Arthur 
was  sent  to  a  private  school.  He  early  showed  great  liking 
for,  and  aptitude  in,  numerical  calculations.  At  the  age  of 
14  he  was  sent  to  King's  College  School,  London;  the  master 
of  which,  having  observed  indications  of  mathematical  genius, 
advised  the  father  to  educate  his  son,  not  for  his  own  business, 
as  he  had  at  first  intended,  but  to  enter  the  University  of 
Cambridge. 

At  the  unusually  early  age  of  17  Cayley  began  residence 
at  Trinity  College,  Cambridge.  As  an  undergraduate  he  had 
generally  the  reputation  of  being  a  mere  mathematician; 
his  chief  diversion  was  novel-reading.  He  was  also  fond  of 
travelling  and  mountain  climbing,  and  was  a  member  of 
the  Alpine  Club.  The  cause  of  the  Analytical  Society  had 
now  triumphed,  and  the  Cambridge  Mathematical  Journal 
had  been  instituted  by  Gregory  and  Leslie  Ellis.  To  this 
journal,  at  the  age  of  twenty,  Cayley  contributed  three 
papers,  on  subjects  which  had  been  suggested  by  reading  the 
Mecanique  analytique  of  Lagrange  and  some  of  the  works  of 
Laplace.  We  have  already  noticed  how  the  works  of  Lagrange 

*  This  Lecture  was  delivered  April  20,  1901. — EDITORS. 
64 


ARTHUR   CAYLEY  65 

and  Laplace  served  to  start  investigation  in  Hamilton  and  Boole. 
Cayley  finished  his  undergraduate  course  by  winning  the  place 
of  Senior  Wrangler,  and  the  first  Smith's  prize.  His  next  step 
was  to  take  the  M.A.  degree,  and  win  a  Fellowship  by  com- 
petitive examination.  He  continued  to  reside  at  Cambridge 
for  four  years;  during  which  time  he  took  some  pupils,  but  his 
main  work  was  the  preparation  of  28  memoirs  to  the  Mathe- 
matical Journal.  On  account  of  the  limited  tenure  of  his 
fellowship  it  was  necessary  to  choose  a  profession;  like  De 
Morgan,  Cayley  chose  the  law,  and  at  25  entered  at  Lincoln's 
Inn,  London.  He  made  a  specialty  of  conveyancing  and  became 
very  skilled  at  the  work;  but  he  regarded  his  legal  occupation 
mainly  as  the  means  of  providing  a  livelihood,  and  he  reserved 
with  jealous  care  a  due  portion  of  his  time  for  mathematical 
research.  It  was  while  he  was  a  pupil  at  the  bar  that  he  went 
over  to  Dublin  for  the  express  purpose  of  hearing  Hamilton's 
lectures  on  Quaternions.  He  sat  alongside  of  Salmon  (now 
provost  of  Trinity  College,  Dublin)  and  the  readers  of  Salmon's 
books  on  Analytical  Geometry  know  how  much  their  author 
was  indebted  to  his  correspondence  with  Cayley  in  the  matter 
of  bringing  his  textbooks  _up  to  date.  His  friend  Sylvester, 
his  senior  by  five  years  at  Cambridge,  was  then  an  actuary, 
resident  in  London;  they  used  to  walk  together  round  the 
courts  of  Lincoln's  Inn,  discussing  the  theory  of  invariants  and 
covariants.  During  this  period  of  his  life,  extending  over  four- 
teen years,  Cayley  produced  between  two  and  three  hundred 
papers. 

At  Cambridge  University  the  ancient  professorship  of  pure 
mathematics  is  denominated  the  Lucasian,  and  is  the  chair 
which  was  occupied  by  Sir  Isaac  Newton.  About  1860  certain 
funds  bequeathed  by  Lady  Sadleir  to  the  University,  having 
become  useless  for  their  original  purpose,  were  employed  to 
establish  another  professorship  of  pure  mathematicas,  called 
the  Sadlerian.  The  duties  of  the  new  professor  were  defined 
to  be  "  to  explain  and  teach  the  principles  of  pure  mathematics 
and  to  apply  himself  to  the  advancement  of  that  science."  To 
this  chair  Cayley  was  elected  when  42  years  old.  He  gave 


65  TEN   BRITISH   MATHEMATICIANS 

up  a  lucrative  practice  for  a  modest  salary;  but  he  never 
regretted  the  exchange,  for  the  chair  at  Cambridge  enabled 
him  to  end  the  divided  allegiance  between  law  and  mathematics, 
and  to  devote  his  energies  to  the  pursuit  which  he  liked  best. 
He  at  once  married  and  settled  down  in  Cambridge.  More 
fortunate  than  Hamilton  in  his  choice,  his  home  life  was  one 
of  great  happiness.  His  friend  and  fellow  investigator,  Sylvester, 
once  remarked  that  Cayley  had  been  much  more  fortunate 
than  himself;  that  they  both  lived  as  bachelors  in  London, 
but  that  Cayley  had  married  and  settled  down  to  a  quiet  and 
peaceful  life  at  Cambridge;  whereas  he  had  never  married, 
and  had  been  fighting  the  world  all  his  days.  The  remark  was 
only  too  true  (as  may  be  seen  in  the  lecture  on  Sylvester). 

At  first  the  teaching  duty  of  the  Sadlerian  professorship  was 
limited  to  a  course  of  lectures  extending  over  one  of  the  terms 
of  the  academic  year;  but  when  the  University  was  reformed 
about  1886,  and  part  of  the  college  funds  applied  to  the  better 
endowment  of  the  University  professors,  the  lectures  were 
extended  over  two  terms.  For  many  years  the  attendance  was 
small,  and  came  almost  entirely  from  those  who  had  finished 
their  career  of  preparation  for  competitive  examinations;  after 
the  reform  the  attendance  numbered  about  fifteen.  The  subject 
lectured  on  was  generally  that  of  the  memoir  on  which  the 
professor  was  for  the  time  engaged. 

The  other  duty  of  the  chair — the  advancement  of  mathe- 
matical science — was  discharged  in  a  handsome  manner  by  the 
long  series  of  memoirs  which  he  published,  ranging  over  every 
department  of  pure  mathematics.  But  it  was  also  discharged 
in  a  much  less  obtrusive  way;  he  became  the  standing  referee 
on  the  merits  of  mathematical  papers  to  many  societies 
both  at  home  and  abroad.  Many  mathematicians,  of  whom 
Sylvester  was  an  example,  find  it  irksome  to  study  what  others 
have  written,  unless,  perchance,  it  is  something  dealing  directly 
with  their  own  line  of  work.  Cayley  was  a  man  of  more  cos- 
mopolitan spirit;  he  had  a  friendly  sympathy  with  other  workers, 
and  especially  with  young  men  making  their  first  adventure  in 
the  field  of  mathematical  research.  Of  referee  work  he  did  an 


ARTHUR   CAYLEY  67 

•  i 

immense  amount;  and  of  his  kindliness  to  young  investigators 
I  can  speak  from  personal  experience.  Several  papers  which 
I  read  before  the  Royal  Society  of  Edinburgh  on  the  Analysis 
of  Relationships  were  referred  to  him,  and  he  recommended 
their  publication.  Soon  after  I  was  invited  by  the  Anthropo- 
logical Society  of  London  to  address  them  on  the  subject, 
and  while  there,  I  attended  a  meeting  of  the  Mathematical 
Society  of  London.  The  room  was  small,  and  some  twelve 
mathematicians  were  assembled  round  a  table,  among  whom 
was  Prof.  Cayley,  as  became  evident  to  me  from  the  proceed- 
ings. At  the  close  of  the  meeting  Cayley  gave  me  a  cordial 
handshake  and  referred  in  the  kindest  terms  to  my  papers 
which  he  had  read.  He  was  then  about  60  years  old,  con- 
siderably bent,  and  not  filling  his  clothes.  What  was  most 
remarkable  about  him  was  the  active  glance  of  his  gray  eyes1 
and  his  peculiar  boyish  smile. 

In  1876  he  published  a  Treatise  on  Elliptic  Functions,  which 
was  his  only  book.  He  took  great  interest  in  the  movement 
for  the  University  education  of  women.  At  Cambridge  the 
women's  colleges  are  Girton  and  Newnham.  In  the  early  days 
of  Girton  College  he  gave  direct  help  in  teaching,  and  for  some 
years  he  was  chairman  of  the  council  of  Newnham  College, 
in  the  progress  of  which  he  took  the  keenest  interest  to  the 
last.  His  mathematical  investigations  did  not  make  him  a 
recluse;  on  the  contrary  he  was  of  great  practical  usefulness, 
especially  from  his  knowledge  of  law,  in  the  administration  of 
the  University. 

In  1872  he  was  made  an  honorary  fellow  of  Trinity  College, 
and  three  years  later  an  ordinary  fellow,  which  meant  stipend 
as  well  as  honor.  About  this  time  his  friends  subscribed  for 
a  presentation  portrait,  which  now  hangs  on  the  side  wall 
of  the  dining  hal"  of  Trinity  College,  next  to  the  portrait  of 
James  Clerk  Maxwell,  while  on  the  end  wall,  behind  the 
high  table,  hang  the  more  ancient  portraits  of  Sir  Isaac  New- 
ton and  Lord  Bacon  of  Verulam.  In  the  portrait  Cayley  is 
represented  as  seated  at  a  desk,  quill  in  hand,  after  the  mode 
in  which  he  used  to  write  out  his  mathematical  investigations. 


68  TEN  BRITISH  MATHEMATICIANS 

The  investigation,  however,  was  all  thought  out  in  his  mind 
before  he  took  up  the  quill. 

Maxwell  was  one  of  the  greatest  electricians  of  the  nine- 
teenth century.  He  was  a  man  of  philosophical  insight  and 
poetical  power,  not  unlike  Hamilton,  but  differing  in  this,  that 
he  was  no  orator.  In  that  respect  he  was  more  like  Gold- 
smith, who  "  could  write  like  an  angel,  but  only  talked  like 
poor  poll."  Maxwell  wrote  an  address  to  the  committee  of 
subscribers  who  had  charge  of  the  Cayley  protrait  fund,  wherein 
the  scientific  poet  with  his  pen  does  greater  honor  to  the  mathe- 
matician than  the  artist,  named  Dickenson,  could  do  with  his 
brush.  Cayley  had  written  on  space  of  n  dimensions,  and  the 
main  point  in  the  address  is  derived  from  the  artist's  business 
of  depicting  on  a  plane  what  exists  in  space: 

O  wretched  race  of  men,  to  space  confined! 

What  honor  can  ye  pay  to  him  whose  mind 

To  that  which  lies  beyond  hath  penetrated? 

The  symbols  he  hath  formed  shall  sound  his  praise, 

And  lead  him  on  through  unimagined  ways 

To  conquests  new,  in  worlds  not  yet  created. 

First,  ye  Determinants,  in  ordered  row 
And  massive  column  ranged,  before  him  go, 
To  form  a  phalanx  for  his  safe  protection. 
Ye  powef s  of  the  nth  root  of  —  i ! 
Around  his  head  in  endless  cycles  run, 
As  unembodied  spirits  of  direction. 

And  you,  ye  undevelopable  scrolls! 

Above  the  host  where  your  emblazoned  rolls, 

Ruled  for  the  record  of  his  bright  inventions. 

Ye  cubic  surfaces!  by  threes  and  nines 

Draw  round  his  camp  your  seven  and  twenty  lines 

The  seal  of  Solomon  in  three  dimensions. 

March  on,  symbolic  host !  with  step  sublime, 
Up  to  the  flaming  bounds  of  Space  and  Time! 
There  pause,  until  by  Dickenson  depicted 
In  two  dimensions,  we  the  form  may  trace 
Of  him  whose  soul,  too  large  for  vulgar  space, 
In  n  dimensions  flourished  unrestricted. 


ARTHUR   CAYLEY  69 

The  verses  refer  to  the  subjects  investigated  in  several  of 
Cayley 's  most  elaborate  memoirs;  such  as,  Chapters  on  the 
Analytical  Geometry  of  n  dimensions;  On  the  theory  of  De- 
terminants; Memoir  on  the  theory  of  Matrices;  Memoirs 
on  skew  surfaces,  otherwise  Scrolls;  On  the  delineation  of  a 
Cubic  Scroll,  etc. 

In  1 88 1  he  received  from  the  Johns  Hopkins  University, 
Baltimore,  where  Sylvester  was  then  professor  of  mathematics, 
an  invitation  to  deliver  a  course  of  lectures.  He  accepted 
the  invitation,  and  lectured  at  Baltimore  during  the  first  five 
months  of  1882  on  the  subject  of  the  Abelian  and  Theta  Functions. 

The  next  year  Cayley  came  prominently  before  the  world, 
as  President  of  the  British  Association  for  the  Advancement 
of  Science.  The  meeting  was  held  at  Southport,  in  the  north 
of  England.  As  the  President's  address  is  one  of  the  great 
popular  events  of  the  meeting,  and  brings  out  an  audience  of 
general  culture,  it  is  usually  made  as  little  technical  as  pos- 
sible. Hamilton  was  the  kind  of  mathematician  to  suit  such 
an  occasion,  but  he  never  got  the  office,  on  account  of  his 
occasional  breaks.  Cayley  had  not  the  oratorical,  the  philo- 
sophical, or  the  poetical  gifts  of  Hamilton,  but  then  he  was  an 
eminently  safe  man.  He  took  for  his  subject  the  Progress  of 
Pure  Mathematics;  and  he  opened  his  address  in  the  following 
na'ive  manner:  "  I  wish  to  speak  to  you  to-night  upon  Mathe- 
matics. I  am  quite  aware  of  the  difficulty  arising  from  the 
abstract  nature  of  my  subject;  and  if,  as  I  fear,  many  or  some 
of  you,  recalling  the  providential  addresses  at  former  meetings, 
should  wish  that  you  were  now  about  to  have  from  a  different 
President  a  discourse  on  a  different  subject,  I  can  very  well 
sympathize  with  you  in  the  feeling.  But  be  that  as  it  may, 
I  think  it  is  more  respectful  to  you  that  I  should  speak  to  you 
upon  and  do  my  best  to  interest  you  in  the  subject  which  has 
occupied  me,  and  in  which  I  am  myself  most  interested.  And 
in  another  point  of  view,  I  think  it  is  right  that  the  address 
of  a  president  should  be  on  his  own  subject,  and  that  different 
subjects  should  be  thus  brought  in  turn  before  the  meetings. 
So  much  the  worse,  it  may  be,  for  a  particular  meeting:  but 


70  TEN   BRITISH  MATHEMATICIANS 

the  reeling  is  the  individual,  which  on  evolution  principles, 
must  be  sacrificed  for  the  development  of  the  race."  I  dare- 
say that  after  this  introduction,  all  the  evolution  philosophers 
listened  to  him  attentively,  whether  they  understood  him  or 
not.  But  Cayley  doubtless  felt  that  he  was  addressing  not 
only  the  popular  audience  then  and  there  before  him,  but  the 
mathematicians  of  distant  places  and  future  times;  for  the 
address  is  a  valuable  historical  review  of  various  mathematical 
theories,  and  is  characterized  by  freshness,  independence  of 
view,  suggestiveness,  and  learning. 

In  1889  the  Cambridge  University  Press  requested  him  to 
prepare  his  mathematical  papers  for  publication  in  a  collected 
form — a  request  which  he  appreciated  very  much.  They  are 
printed  in  magnificent  quarto  volumes,  of  which  seven  appeared 
under  his  own  editorship.  While  editing  these  volumes,  he 
was  suffering  from  a  painful  internal  malady,  to  which  he 
succumbed  on  January  26,  1895,  m  the  74-th  year  of  his  age. 
When  the  funeral  took  place,  a  great  assemblage  met  in  Trinity 
Chapel,  comprising  members  of  the  University,  official  rep- 
resentatives of  Russia  and  America,  and  many  of  the  most 
illustrious  philosophers  of  Great  Britain. 

The  remainder  of  his  papers  were  edited  by  Prof.  Forsyth, 
his  successor  in  the  Sadlerian  chair.  The  Collected  Mathe- 
matical papers  number  thirteen  quarto  volumes,  and  con- 
tain 967  papers.  His  writings  are  his  best  monument,  and 
certainly  no  mathematician  has  ever  had  his  monument  in 
grander  style.  De  Morgan's  works  would  be  more  extensive, 
and  much  more  useful,  but  he  did  not  have  behind  him  a 
University  Press.  As  regards  fads,  Cayley  retained  to  the 
last  his  fondness  for  novel-reading  and  for  travelling.  He 
also  took  special  pleasure  in  paintings  and  architecture,  and 
he  practised  water-color  painting,  which  he  found  useful  some- 
times in  making  mathematical  diagrams. 

To  the  third  edition  of  Tait's  Elementary  Treatise  on  Qua- 
ternions, Cayley  contributed  a  chapter  entitled  "  Sketch  of 
the  analytical  theory  of  quaternions."  In  it  the  V  — i  re- 
appears in  all  its  glory,  and  in  entire,  so  it  is  said,  independence 


ARTHUR  CAYLEY  71 

of  i,j,  k.  The  remarkable  thing  is  that  Hamilton  started  with 
a  quaternion  theory  of  analysis,  and  that  Cayley  should  present 
instead  an  analytical  theory  of  quaternions.  I  daresay  that 
Prof.  Tait  was  sorry  that  he  allowed  the  chapter  to  enter 
his  book,  for  in  1894  there  arose  a  brisk  discussion  between 
himself  and  Cayley  on  "  Coordinates  versus  Quaternions," 
the  record  of  which  is  printed  in  the  Proceedings  of  the  Royal 
Society  of  Edinburgh.  Cayley  maintained  the  position  that 
while  coordinates  are  applicable  to  the  whole  science  of  geom- 
etry and  are  the  natural  and  appropriate  basis  and  method 
in  the  science,  quaternions  seemed  a  particular  and  very  arti- 
ficial method  for  treating  such  parts  of  the  science  of  three- 
dimensional  geometry  as  are  most  naturally  discussed  by  means 
of  the  rectangular  coordinates  x,  y,  z.  In  the  course  of  his 
paper  Cayley  says:  "  I  have  the  highest  admiration  for  the 
notion  of  a  quaternion;  but,  as  I  consider  the  full  moon  far 
more  beautiful  than  any  moonlit  view,  so  I  regard  the  notion 
of  a  quaternion  as  far  more  beautiful  than  any  of  its  applica- 
tions. As  another  illustration,  I  compare  a  quaternion  formula 
to  a  pocket-map — a  capital  thing  to  put  in  one's  pocket,  but 
which  for  use  must  be  unfolded:  the  formula,  to  be  under- - 
stood,  must  be  translated  into  coordinates."  He  goes  on 
to  say,  "  I  remark  that  the  imaginary  of  ordinary  algebra — 
for  distinction  call  this  6 — has  no  relation  whatever  to  the 
quaternion  symbols  i,  j,  k]  in  fact,  in  the  general  point  of 
view,  all  the  quantities  which  present  themselves,  are,  or  may 
be,  complex  values  a+6b,  or  in  other  words,  say  that  a  scalar 
quantity  is  in  general  of  the  form  a-}- Ob.  Thus  quaternions  do 
not  properly  present  themselves  in  plane  or  two-dimensional 
geometry  at  all;  but  they  belong  essentially  to  solid  or  three- 
dimensional  geometry,  and  they  are  most  naturally  applicable 
to  the  class  of  problems  which  in  coordinates  are  dealt  with 
by  means  of  the  three  rectangular  coordinates  x,  y,  z." 

To  the  pocketbook  illustration  it  may  be  replied  that  a  set 
of  coordinates  is  an  immense  wall  map,  which  you  cannot  carry 
about,  even  though  you  should  roll  it  up,  and  therefore  is 
useless  for  many  important  purposes.  In  reply  to  the  argu- 


72  TEN  BRITISH  MATHEMATICIANS 

ments,  it  may  be  said,  first,  V  — i  has  a  relation  to  the  symbols 
i,  j,  k,  for  each  of  these  can  be  analyzed  into  a  unit  axis  mul- 
tiplied by  V  — i;  second,  as  regards  plane  geometry,  the 
ordinary  form  of  complex  quantity  is  a  degraded  form  of  the 
quaternion  in  which  the  constant  axis  of  the  plane  is  left  un- 
specified. Cayley  took  his  illustrations  from  his  experience  as 
a  traveller.  Tait  brought  forward  an  illustration  from  which 
you  might  imagine  he  had  visited  the  Bethlehem  Iron 
Works,  and  hunted  tigers  in  India.  He  says,  "  A  much  more 
natural  and  adequate  comparison  would,  it  seems  to  me, 
liken  Coordinate  Geometry  to  a  steam-hammer,  which  an 
expert  may  employ  on  any  destructive  or  constructive  work  of 
one  general  kind,  say  the  cracking  of  an  eggshell,  or  the  weld- 
ing of  an  anchor.  But  you  must  have  your  expert  to  manage 
it,  for  without  him  it  is  useless.  He  has  to  toil  amid  the  heat, 
smoke,  grime,  grease,  and  perpetual  din  of  the  suffocating 
engine-room.  The  work  has  to  be  brought  to  the  hammer, 
for  it  cannot  usually  be  taken  to  its  work.  And  it  is  not  in 
general,  transferable;  for  each  expert,  as  a  rule,  knows,  fully 
and  confidently,  the  working  details  of  his  own  weapon  only. 
Quaternions,  on  the  other  hand,  are  like  the  elephant's  trunk, 
ready  at  any  moment  for  anything,  be  it  to  pick  up  a  crumb 
or  a  field-gun,  to  strangle  a  tiger,  or  uproot  a  tree;  portable 
in  the  extreme,  applicable  anywhere — alike  in  the  trackless 
jungle  and  in  the  barrack  square — directed  by  a  little  native 
who  requires  no  special  skill  or  training,  and  who  can  be  trans- 
ferred from  one  elephant  to  another  without  much  hesitation. 
Surely  this,  which  adapts  itself  to  its  work,  is  the  grander 
instrument.  But  then,  it  is  the  natural,  the  other,  the  arti- 
ficial one." 

The  reply  which  Tait  makes,  so  far  as  it  is  an  argument, 
is:  There  are  two  systems  of  quaternions,  the  i,  j,  k  one,  and 
another  one  which  Hamilton  developed  from  it;  Cayley  knows 
the  first  only,  he  himself  knows  the  second;  the  former  is  an 
intensely  artificial  system  of  imaginaries,  the  latter  is  the 
natural  organ  of  expression  for  quantities  in  space.  Should  a 
fourth  edition  of  his  Elementary  Treatise  be  called  for  i}j}  k  will 


ARTHUR   CAYLEY^  73 

disappear  from  it,  excepting  in  Cayley's  chapter,  should  it 
be  retained.  Tait  thus  describes  the  first  system :  "Hamilton's 
extraordinary  Preface  to  his  first  great  book  shows  how  from 
Double  Algebras,  through  Triplets,  Triads,  and  Sets,  he  finally 
reached  Quaternions.  This  was  the  genesis  of  the  Quaternions 
of  the  forties,  and  the  creature  thus  produced  is  still  essentially 
the  Quaternion  of  Prof.  Cayley.  It  is  a  magnificent  analytical 
conception;  but  it  is  nothing  more  than  the  full  development 
of  the  system  ojf  imaginaries  i,  j,  k ;  defined  by  the  equations, 

•O  *O  7O  •  *T 

*    =J     =k    =1]K=  —I, 

with  the  associative,  but  not  the  commutative,  law  for  the 
factors.  The  novel  and  splendid  points  in  it  were  the  treat- 
ment of  all  directions  in  space  as  essentially  alike  in  character, 
and  the  recognition  of  the  unit  vector's  claim  to  rank  also  as 
a  quadrantal  versor.  These  were  indeed  inventions  of  the 
first  magnitude,  and  of  vast  importance.  And  here  I  thor- 
oughly agree  with  Prof.  Cayley  in  his  admiration.  Considered 
as  an  analytical  system,  based  throughout  on  pure  imaginaries, 
the  Quaternion  method  is  elegant  in  the  extreme.  But,  unless 
it  had  been  also  something  more,  something  very  different 
and  much  higher  in  the  scale  of  development,  I  should  have 
been  content  to  admire  it: — and  to  pass  it  by." 

From  '  the  most  intensely  artificial  of  systems,  arose,  as 
if  by  magic,  an  absolutely  natural  one  "  which  Tait  thus  fur- 
ther describes.  "  To  me  Quaternions  are  primarily  a  Mode 
of  Representation: — immensely  superior  to,  but  of  essentially 
the  same  kind  of  usefulness  as,  a  diagram  or  a  model.  They 
are,  virtually,  the  thing  represented;  and  are  thus  antecedent 
to,  and  independent  of,  coordinates;  giving,  in  general,  all 
the  main  relations,  in  the  problem  to  which  they  are  applied, 
without  the  necessity  of  appealing  to  coordinates  at  all.  Co- 
ordinates may,  however,  easily  be  read  into  them: — when  any- 
thing (such  as  metr  cal  or  numerical  detail)  is  to  be  gained 
thereby.  Quaternions,  in  a  word,  exist  in  space,  and  we  have 
only  to  recognize  them: — but  we  have  to  invent  or  imagine 
coordinates  of  all  kinds." 


74  TEN  BRITISH  MATHEMATICIANS 

To  meet  the  objection  why  Hamilton  did  not  throw  i,  j,  k 
overboard,  and  expound  the  developed  system,  Tait  says: 
"  Most  unfortunately,  alike  for  himself  and  for  his  grand  con- 
ception, Hamilton's  nerve  failed  him  in  the  composition  of 
his  first  great  volume.  Had  he  then  renounced,  for  ever,  all 
dealings  with  i,  j,  k,  his  triumph  would  have  been  complete. 
He  spared  Agog,  and  the  best  of  the  sheep,  and  did  not  utterly 
destroy  them.  He  had  a  paternal  fondness  for  i,j,  k',  perhaps 
also  a  not  unnatural  liking  for  a  meretricious  title  such  as 
the  mysterious  word  Quaternion;  and,  above  all,  he  had  an 
earnest  desire  to  make  the  utmost  return  in  his  power  for  the 
liberality  shown  him  by  the  authorities  of  Trinity  College, 
Dublin.  He  had  fully  recognized,  and  proved  to  others,  that 
his  i,  j,  k,  were  mere  excrescences  and  blots  on  his  improved 
method: — but  he  unfortunately  considered  that  their  continued 
(if  only  partial)  recognition  was  indispensable  to  the  reception 
of  his  method  by  a  world  steeped  in — Cartesianism !  Through 
the  whole  compass  of  each  of  his  tremendous  volumes  one  can 
find  traces  of  his  desire  to  avoid  even  an  allusion  to  i,  j,  k, 
and  along  with  them,  his  sorrowful  conviction  that,  should  he 
do  so,  he  would  be  left  without  a  single  reader." 

To  Cayley's  presidential  address  we  are  indebted  for  in- 
formation about  the  view  which  he  took  of  the  foundations  of 
exact  science,  and  the  philosophy  which  commended  itself  to 
his  mind.  He  quoted  Plato  and  Kant  with  approval,  J.  S. 
Mill  with  faint  praise.  Although  he  threw  a  sop  to  the  empiri- 
cal philosophers  at  the  beginning  of  his  address,  he  gave  them 
something  to  think  of  before  he  finished. 

He  first  of  all  remarks  that  the  connection  of  arithmetic 
and  algebra  with  the  notion  of  time  is  far  less  obvious  than 
that  of  geometry  with  the  notion  of  space;  in  which  he,  of 
course,  made  a  hit  at  Hamilton's  theory  of  Algebra  as  the 
science  of  pure  time.  Further  on  he  discusses  the  theory 
directly,  and  concludes  as  follows:  "  Hamilton  uses  the  term 
algebra  in  a  very  wide  sense,  but  whatever  else  he  includes 
under  it,  he  includes  all  that  in  contradistinction  to  the  Dif- 
ferential Calculus  woukfbe  called  algebra.  Using  the  word 


ARTHUR  CAYLEY  75 

in  this  restricted  sense,  I  cannot  myself  recognize  the  con- 
nection of  algebra  with  the  notion  of  time;  granting  that  the 
notion  of  continuous  progression  presents  itself  and  is  of  im- 
portance, I  do  not  see  that  it  is  in  anywise  the  fundamental 
notion  of  the  science.  And  still  less  can  I  appreciate  the  manner 
in  which  the  author  connects  with  the  notion  of  time  his  alge- 
braical couple,  or  imaginary  magnitude,  a-\-b\/  —  i."  So  you 
will  observe  that  doctors  differ — Tait  and  Cayley — about  the 
soundness  of  Hamilton's  theory  of  couples.  But  it  can  be  shown 
that  a  couple  may  not  only  be  represented  on  a  straight  line, 
but  actually  means  a  portion  of  a  straight  line;  and  as  a  line 
is  unidimensional,  this  favors  the  truth  of  Hamilton's  theory. 

As  to  the  nature  of  mathematical  science  Cayley  quoted 
with  approval  from  an  address  of  Hamilton's: 

"  These  purely  mathematical  sciences  of  algebra  and  geom- 
etry are  sciences  of  the  pure  reason,  deriving  no  weight  and 
no  assistance  from  experiment,  and  isolated  or  at  least  isolable 
from  all  outward  and  accidental  phenomena.  The  idea  of  order 
with  its  subordinate  ideas  of  number  and  figure,  we  must  not 
call  innate  ideas,  if  that  phrase  be  defined  to  imply  that  all 
men  must  possess  them  with  equal  clearness  and  fulness;  they 
are,  however,  ideas  which  seem  to  be  so  far  born  with  us  that 
the  possession  of  them  in  any  conceivable  degree  is  only  the 
development  of  our  original  powers,  the  unfolding  of  our  proper 
humanity." 

It  is  the  aim  of  the  evolution  philosopher  to  reduce  all 
knowledge  to  the  empirical  status;  the  only  intuition  he  grants 
is  a  kind  of  instinct  formed  by  the  experience  of  ancestors  and 
transmitted  cumulatively  by  heredity.  Cayley  first  takes  him  up 
on  the  subject  of  arithmetic:  "  Whatever  difficulty  be  raisable 
as  to  geometry,  it  seems  to  me  that  no  similar  difficulty  applies 
to  arithmetic;  mathematician,  or  not,  we  have  each  of  us,  in 
its  most  abstract  form,  the  idea  of  number;  we  can  each  of  us 
appreciate  the  truth  of  a  proposition  in  numbers;  and  we  cannot 
but  see  that  a  truth  in  regard  to  numbers  is  something  different 
in  kind  from  an  experimental  truth  generalized  from  experience. 
Compare,  for  instance,  the  proposition,  that  the  sun,  having 


76  TEN   BRITISH  MATHEMATICIANS 

already  risen  so  many  times,  will  rise  to-morrow,  and  the  next 
day,  and  the  day  after  that,  and  so  on;  and  the  proposition 
that  even  and  odd  numbers  succeed  each  other  alternately  ad 
infinitum;  the  latter  at  least  seems  to  have  the  characters  of 
universality  and  necessity.  Or  again,  suppose  a  proposition 
observed  to  hold  good  for  a  long  series  of  numbers,  one  thousand 
numbers,  two  thousand  numbers,  as  the  case  may  be:  this  is 
not  only  no  proof,  but  it  is  absolutely  no  evidence,  that  the 
proposition  is  a  true  proposition,  holding  good  for  all  numbers 
whatever;  there  are  in  the  Theory  of  Numbers  very  remark- 
able instances  of  propositions  observed  to  hold  good  for  very 
long  series  of  numbers  which  are  nevertheless  untrue." 

Then  he  takes  him  up  on  the  subject  of  geometry,  where 
the  empiricist  rather  boasts  of  his  success.  "  It  is  well  known 
that  Euclid's  twelfth  axiom,  even  in  Playfair's  form  of  it,  has 
been  considered  as  needing  demonstration;  and  that  Lobat- 
schewsky  constructed  a  perfectly  consistent  theory,  wherein 
this  axiom  was  assumed  not  to  hold  good,  or  say  a  system  of 
non-Euclidean  plane  geometry.  My  own  view  is  that  Euclid's 
twelfth  axiom  in  Playfair's  form  of  it  does  not  need  demonstra- 
tion, but  is  part  of  our  notion  of  space,  of  the  physical  space  of 
our  experience — the  space,  that  is,  which  we  become  acquainted 
with  by  experience,  but  which  is  the  representation  lying  at  the 
foundation  of  all  external  experience.  Riemann's  view  before 
referred  to  may  I  think  be  said  to  be  that,  having  in  intellectu 
a  more  general  notion  of  space  (in  fact  a  notion  of  non-Euclidean 
space),  we  learn  by  experience  that  space  (the  physical  space 
of  our  experience)  is,  if  not  exactly,  at  least  to  the  highest  degree 
of  approximation,  Euclidean  space.  But  suppose  the  physical 
space  of  our  experience  to  be  thus  only  approximately  Euclidean 
space,  what  is  the  consequence  which  follows?  Not  that  the  prop- 
ositions of  geometry  are  only  approximately  true,  but  that  they 
remain  absolutely  true  in  regard  to  that  Euclidean  space  which 
has  been  so  long  regarded  as  being  the  physical  space  of  our 
experience." 

In  his  address  he  remarks  that  the  fundamental  notion 
which  underlies  and  pervades  the  whole  of  modern  analysis  and 


ARTHUR  CAYLEY  77 

geometry  is  that  of  imaginary  magnitude  in  analysis  and  of 
imaginary  space  (or  space  as  a  locus  in  quo  of  imaginary  points 
and  figures)  in  geometry.  In  the  case  of  two  given  curves 
there  are  two  equations  satisfied  by  the  coordinates  (x,  y)  of  the 
several  points  of  intersection,  and  these  give  rise  to  an  equation 
of  a  certain  order  for  the  coordinate  x  or  y  of  a  point  of  inter- 
section. In  the  case  of  a  straight  line  and  a  circle  this  is  a 
quadratic  equation;  it  has  two  roots  real  or  imaginary.  There 
are  thus  two  values,  say  of  x,  and  to  each  of  these  corresponds 
a  single  value  of  y.  There  are  therefore  two  points  of  inter- 
section, viz.,  a  straight  line  and  a  circle  intersect  always  in 
two  points,  real  or  imaginary.  It  is  in  this  way  we  are  led 
analytically  to  the  notion  of  imaginary  points  in  geometry. 
He  asks,  What  is  an  imaginary  point?  Is  there  in  a  plane  a 
point  the  coordinates  of  which  have  given  imaginary  values? 
He  seems  to  say  No,  and  to  fall  back  on  the  notion  of  an  imagi- 
nary space  as  the  locus  in  quo  of  the  imaginary  point. 


WILLIAM  KINGDON  CLIFFORD* 
(1845-1879) 

WILLIAM  KINGDON  CLIFFORD  was  born  at  Exeter,  England, 
May  4,  1845.  His  father  was  a  well-known  and  active  citizen 
and  filled  the  honorary  office  of  justice  of  the  peace;  his  mother 
died  while  he  was  still  young.  It  is  believed  that  Clifford 
inherited  from  his  mother  not  only  some  of  his  genius,  but  a 
weakness  in  his  physical  constitution.  He  received  his  ele- 
mentary education  at  a  private  school  in  Exeter,  where  examina- 
tions were  annually  held  by  the  Board  of  Local  Examinations 
of  the  Universities  of  Oxford  and  Cambridge;  at  these  examina- 
tions Clifford  gained  numerous  distinctions  in  widely  different 
subjects.  When  fifteen  years  old  he  was  sent  to  King's  College, 
London,  where  he  not  only  demonstrated  his  peculiar  mathe- 
matical abilities,  but  also  gamed  distinction  in  classics  and 
English  literature. 

When  eighteen,  he  entered  Trinity  College,  Cambridge;  the 
college  of  Peacock,  De  Morgan,  and  Cayley.  He  already  had 
the  reputation  of  possessing  extraordinary  mathematical  powers; 
and  he  was  eccentric  in  appearance,  habits  and  opinions.  He 
was  reported  to  be  an  ardent  High  Churchman,  which  was  then 
a  more  remarkable  thing  at  Cambridge  than  it  is  now.  His 
undergraduate  career  was  distinguished  by  eminence  in  mathe- 
matics, English  literature  and  gymnastics.  One  who  was  his  com- 
panion in  gymnastics  wrote:  "  His  neatness  and  dexterity  were 
unusually  great,  but  the  most  remarkable  thing  was  his  great 
strength  as  compared  with  his  weight,  as  shown  in  some  exercises. 
At  one  time  he  would  pull  up  on  the  bar  with  either  hand,  which 
is  well  known  to  be  one  of  the  greatest  feats  of  strength.  His 
nerve  at  dangerous  heights  was  extraordinary."  In  his  third 

*  This  Lecture  was  delivered  April  23,  1901. — EDITORS. 
78 


WILLIAM   KINGDON   CLIFFORD  79 

year  he  won  the  prize  awarded  by  Trinity  College  for  decla- 
mation, his  subject  being  Sir  Walter  Raleigh;  as  a  consequence 
he  was  called  on  to  deliver  the  annual  oration  at  the  next  Com- 
memoration of  Benefactors  of  the  College.  He  chose  for  his 
subject,  Dr.  Whewell,  Master  of  the  College,  eminent  for  his 
philosophical  and  scientific  attainments,  whose  death  had 
occurred  but  recently.  He  treated  it  in  an  original  and  un- 
expected manner;  Dr.  WhewelPs  claim  to  admiration  and 
emulation  being  put  on  the  ground  of  his  intellectual  life  exem- 
plifying in  an  eminent  degree  the  active  and  creating  faculty. 
"  Thought  is  powerless,  except  it  make  something  outside  of 
itself;  the  thought  which  conquers  the  world  is  not  contemplative 
but  active.  And  it  is  this  that  I  am  asking  you  to  worship 
to-day." 

To  obtain  high  honors  in  the  Mathematical  Tripos,  a  student 
must  put  himself  in  special  training  under  a  mathematican, 
technically  called  a  coach,  who  is  not  one  of  the  regular  college 
instructors,  nor  one  of  the  University  professors,  but  simply 
makes  a  private  business  of  training  men  to  pass  that  par- 
ticular examination.  Skill  consists  in  the  rate  at  which  one  can 
solve  and  more  especially  write  out  the  solution  of  problems. 
It  is  excellent  training  of  a  kind,  but  there  is  no  time  for  study- 
ing fundamental  principles,  still  less  for  making  any  philosoph- 
ical investigations.  Mathematical  insight  is  something  higher 
than  skill  in  solving  problems;  consequently  the  senior  wrangler 
has  not  always  turned  out  the  most  distinguished  mathematician 
in  after  life.  We  have  seen  that  De  Morgan  was  fourth  wrangler. 
Clifford  also  could  not  be  kept  to  the  dust  of  the  race-course; 
but  such  was  his  innate  mathematical  insight  that  he  came 
out  second  wrangler.  Other  instances  of  the  second  wrangler 
turning  out  the  better  mathematician  are  Whewell,  Sylvester, 
Kelvin,  Maxwell. 

In  1868,  when  he  was  23  years  old,  he  was  elected  a  Fellow 
of  his  College;  and  while  a  resident  fellow,  he  took  part  in  the 
eclipse  expedition  of  1870  to  Italy,  and  passed  through  the 
experience  of  a  shipwreck  near  Catania  on  the  coast  of  the 
island  of  Sicily.  In  1871  he  was  appointed  professor  of  Ap- 


80  TEN  BRITISH  MATHEMATICIANS 

plied  Mathematics  and  Mechanics  in  University  College, 
London;  De  Morgan's  college,  but  not  De  Morgan's  chair. 
Henceforth  University  College  was  the  centre  of  his  labors. 

He  was  now  urged  by  friends  to  seek  admission  into 
the  Royal  Society  of  London.  This  is  the  ancient  scientific 
society  of  England,  founded  in  the  time  of  Charles  II,  and 
numbering  among  its  first  presidents  Sir  Isaac  Newton.  About 
the  middle  of  the  nineteenth  century  the  admission  of  new 
members  was  restricted  to  fifteen  each  year;  and  from  appli- 
cations the  Council  recommends  fifteen  names  which  are  posted 
up,  and  subsequently  balloted  for  by  the  Fellows.  Hamilton 
and  De  Morgan  never  applied.  Clifford  did  not  apply  imme- 
diately, but  he  became  a  Fellow  a  few  years  later.  He  joined 
the  London  Mathematical  Society — for  it  met  in  University 
College — and  he  became  one  of  its  leading  spirits.  Another 
metropolitan  Society  in  which  he  took  much  interest  was  the 
Metaphysical  Society;  like  Hamilton,  De  Morgan,  and  Boole, 
Clifford  was  a  scientific  philosopher. 

In  1875  Clifford  married;  the  lady  was  Lucy,  daughter  of 
Mr.  John  Lane,  formerly  of  Barbadoes.  His  home  in  London 
became  the  meeting-point  of  a  numerous  body  of  friends,  in 
which  almost  every  possible  variety  of  taste  and  opinion  was 
represented,  and  many  of  whom  had  nothing  else  in  common. 
He  took  a  special  delight  in  amusing  children,  and  for  their 
entertainment  wrote  a  collection  of  fairy  tales  called  The  Little 
People.  In  this  respect  he  was  like  a  contemporary  mathe- 
matician, Mr.  Dodgson — "  Lewis  Carroll  " — the  author  of 
Alice  in  Wonderland.  A  children's  party  was  one  of  Clifford's 
greatest  pleasures.  At  one  such  party  he  kept  a  waxwork 
show,  children  doing  duty  for  the  figures;  but  I  daresay  he 
drew  the  line  at  walking  on  all  fours,  as  Mr.  Dodgson  was  ac- 
customed to  do.  A  children's  party  was  to  be  held  in  a  house 
in  London  and  it  happened  that  there  was  a  party  of  adults 
held  simultaneously  in  the  neighboring  house;  to  give  the 
children  a  surprise  Dodgson  resolved  to  walk  in  on  all  fours; 
unfortunately  he  crawled  into  the  parlor  of  the  wrong  house! 

Clifford  possessed  unsurpassed  power  as  a  teacher.     Mr. 


WILLIAM  KINGDON   CLIFFORD  81 

Pollock,  a  fellow  student,  gives  an  instance  of  Clifford's  theory 
of  what  teaching  ought  to  be,  and  his  constant  way  of  carrying 
it  out  in  his  discourses  and  conversations  on  mathematical 
and  scientific  subjects.  "  In  the  analytical  treatment  of  statics 
there  occurs  a  proposition  called  Ivory's  Theorem  concerning 
the  attractions  of  an  ellipsoid.  The  textbooks  demonstrate 
it  by  a  formidable  apparatus  of  coordinates  and  integrals, 
such  as  we  were  wont  to  call  a  grind.  On  a  certain  day  in  the 
Long  Vacation  of  1866,  which  Clifford  and  I  spent  at  Cambridge, 
I  was  not  a  little  exercised  by  the  theorem  in  question,  as  I 
suppose  many  students  have  been  before  and  since.  The  chain 
of  symbolic  proof  seemed  artificial  and  dead;  it  compelled  the 
understanding,  but  failed  to  satisfy  the  reason.  After  reading 
and  learning  the  proposition  one  still  failed  to  see  what  it  was 
all  about.  Being  out  for  a  walk  with  Clifford,  I  opened  my 
perplexities  to  him;  I  think  that  I  can  recall  the  very  spot. 
What  he  said  I  do  not  remember  in  detail;  which  is  not  sur- 
prising, as  I  have  had  no  occasion  to  remember  anything  about 
Ivory's  Theorem  these  twelve  years.  But  I  know  that  as  he 
spoke  he  appeared  not  to  be  working  out  a  question,  but 
simply  telling  what  he  saw.  Without  any  diagram  or  symbolic 
aid  he  described  the  geometrical  conditions  on  which  the 
solution  depended,  and  they  seemed  to  stand  out  visibly  in 
space.  There  were  no  longer  consequences  to  be  deduced, 
but  real  and  evident  facts  which  only  required  to  be  seen." 

Clifford  inherited  a  constitution  in  which  nervous  energy 
and  physical  strength  were  unequally  balanced.  It  was  in 
his  case  specially  necessary  to  take  good  care  of  his  health, 
but  he  did  the  opposite;  he  would  frequently  sit  up  most  of 
the  night  working  or  talking.  Like  Hamilton  he  would  work 
twelve  hours  on  a  stretch;  but,  unlike  Hamilton,  he  had 
laborious  professional  duties  demanding  his  personal  attention 
at  the  same  time.  The  consequence  was  that  five  years  after 
his  appointment  to  the  chair  in  University  College,  his  health 
broke  down;  indications  of  pulmonary  disease  appeared.  To 
recruit  his  health  he  spent  six  months  in  Algeria  and  Spain, 
and  came  back  to  his  professional  duties  again.  A  year  and 


82  TEN  BRITISH  MATHEMAIICIANS 

a  half  later  his  health  broke  down  a  second  time,  and  he  was 
obliged  to  leave  again  for  the  shores  of  the  Mediterranean. 
In  the  fall  of  1878  he  returned  to  England  for  the  last  time, 
when,  the  winter  came  he  left  for  the  Island  of  Madeira;  all 
hope  of  recovery  was  gone;  he  died  March  3,  1879  in  the  34th 
year  of  his  age. 

On  the  title  page  of  the  volume  containing  his  collected 
mathematical  papers  I  find  a  quotation,  "If  he  had  lived  we 
might  have  known  something."  Such  is  the  feeling  one  has 
when  one  looks  at  his  published  works  and  thinks  of  the  short- 
ness of  his  life.  In  his  lifetime  there  appeared  Elements  of  Dy- 
namic, Part  /.  Posthumously  there  have  appeared  Elements  of 
Dynamic,  Part  II;  Collected  Mathematical  Papers;  Lectures  and 
Essays;  Seeing  and  Thinking;  Common  Sense  of  the  Exact 
Sciences.  The  manuscript  of  the  last  book  was  left  in  a  very 
incomplete  state,  but  the  design  was  filled  up  and  completed 
by  two  other  mathematicians. 

In  a  former  lecture  I  had  occasion  to  remark  on  the  relation 
of  Mathematics  to  Poetry — on  the  fact  that  in  mathematical 
investigation  there  is  needed  a  higher  power  of  imagination 
akin  to  the  creative  instinct  of  the  poet.  The  matter  is  dis- 
cussed by  Clifford  in  a  discourse  on  "  Some  of  the  conditions 
of  mental  development,"  which  he  delivered  at  the  Royal 
Institution  in  1868  when  he  was  23  years  of  age.  This  insti- 
tution was  founded  by  Count  Rumford,  an  American,  and  is 
located  in  London.  There  are  Professorships  of  Chemistry, 
Physics,  and  Physiology;  its  professors  have  included  Davey, 
Faraday,  Young,  Tyndall,  Rayleigh,  Dewar.  Their  duties  are 
not  to  teach  the  elements  of  their  science  to  regular  students, 
but  to  make  investigations,  and  to  lecture  to  the  members 
of  the  institution,  who  are  in  general  wealthy  and  titled  people. 

In  this  discourse  Clifford  said  "  Men  of  science  have  to 
deal  with  extremely  abstract  and  general  conceptions.  By 
constant  use  and  familiarity,  these,  and  the  relations  between 
them,  become  just  as  real  and  external  a3  the  ordinary  objects 
of  experience,  and  the  perception  of  new  relations  among  them 
is  so  rapid,  the  correspondence  of  the  mind  to  external  circum- 


WILLIAM   KINGDON  CLIFFORD  83 

stances  so  great,  that  a  real  scientific  sense  is  developed,  by 
which  things  are  perceived  as  immediately  and  truly  as  I  see 
you  now.  Poets  and  painters  and  musicians  also  are  so  accus- 
tomed to  put  outside  of  them  the  idea  of  beauty,  that  it  becomes 
a  real  external  existence,  a  thing  which  they  see  with  spiritual 
eyes  and  then  describe  to  you,  but  by  no  means  create,  any 
more  than  we  seem  to  create  the  ideas  of  table  and  forms  and 
light,  which  we  put  together  long  ago.  There  is  no  scientific 
discoverer,  no  poet,  no  painter,  no  musician,  who  will  not  tell 
you  that  he  found  ready  made  his  discovery  or  poem  or  picture — 
that  it  came  to  him  from  outside,  and  that  he  did  not  con- 
sciously create  it  from  within.  And  there  is  reason  to  think 
that  these  senses  or  insights  are  things  which  actually  increase 
among  mankind.  It  is  certain,  at  least,  that  the  scientific  sense 
is  immensely  more  developed  now  than  it  was  three  hundred 
years  ago ;  and  though  it  may  be  impossible  to  find  any  absolute 
standard  of  art,  yet  it  is  acknowledged  that  a  number  of  minds 
which  are  subject  to  artistic  training  will  tend  to  arrange 
themselves  under  certain  great  groups  and  that  the  members 
of  each  group  will  give  an  independent  and  yet  consentient 
testimony  about  artistic  questions.  And  this  arrangement 
into  schools,  and  the  definiteness  of  the  conclusions  reached  in 
each,  are  on  the  increase,  so  that  here,  it  would  seem,  are  actually 
two  new  senses,  the  scientific  and  the  artistic,  which  the  mind 
is  now  in  the  process  of  forming  for  itself." 

Clifford  himself  wrote  a  good  many  poems,  but  only  a  few 
have  been  published.  The  following  verses  were  sent  to  George 
Eliot,  the  novelist,  with  a  presentation  copy  of  The  Little  People: 

Baby  drew  a  little  house, 

Drew  it  all  askew; 
Mother  saw  the  crooked  door 

And  the  window  too. 

Mother  heart,  whose  wide  embrace 

Holds  the  hearts  of  men, 
Grows  with  all  our  growing  hopes, 

Gives  them  birth  again, 


84  TEN  BRITISH   MATHEMATICIANS 

Listen  to  this  baby-talk: 
'Tisn't  wise  or  clear; 
But  what  baby-sense  it  has 
Is  for  you  to  hear. 

An  amusement  in  which  Clifford  took  pleasure  even  in  his 
maturer  years  was  the  flying  of  kites.  He  made  some  mathe- 
matical investigations  in  the  subject,  anticipating,  as  it  were,  the 
interest  which  has  been  taken  in  more  recent  years  in  the  subject 
of  motion  through  the  atmosphere.  Clifford  formed  a  project 
of  writing  a  series  of  textbooks  on  Mathematics  beginning  at 
the  very  commencement  of  each  subject  and  carrying  it  on 
rapidly  to  the  most  advanced  stages.  He  began  with  the 
Elements  of  Dynamic,  of  which  three  books  were  printed  in  his 
lifetime,  and  a  fourth  book,  in  a  supplementary  volume,  after 
his  death.  The  work  is  unique  for  the  clear  ideas  given  of  the 
science;  ideas  and  principles  are  more  prominent  than  symbols 
and  formulae.  He  takes  such  familiar  words  as  spin,  twist, 
squirt,  whirl,  and  gives  them  an  exact  meaning.  The  book  is 
an  example  of  what  he  meant  by  scientific  insight,  and  from  its 
excellence  we  can  imagine  what  the  complete  series  of  text- 
books would  have  been. 

In  Clifford's  lifetime  it  was  said  in  England  that  he  was  the 
only  mathematician  who  could  discourse  on  mathematics  to 
an  audience  composed  of  people  of  general  culture  and  make 
them  think  that  they  understood  the  subject.  In  1872  he  was 
invited  to  deliver  an  evening  lecture  before  the  members  of  the 
British  Association,  at  Brighton;  he  chose  for  his  subject  "  The 
aims  and  instruments  of  scientific  thought."  The  main  theses 
of  the  lecture  are  First,  that  scientific  thought  is  the  application 
of  past  experience  to  new  circumstances  by  means  of  an  observed 
order  of  events.  Second,  this  order  of  events  is  not  th  oreti  ally 
or  absolutely  exact,  but  only  exa,ct  enough  to  correct  experi- 
ments by.  As  an  instance  of  what  is,  and  what  is  not  scientific 
thought,  he  takes  the  phenomenon  of  double  ref  actron.  "  A 
mineralogist,  by  measuring  the  angles  of  a  crystal,  can  tell  you 
whether  or  no  it  possesses  the  property  of  double  refraction 
without  looking  through  it.  He  requires  no  scientific  thought 


WILLIAM   KINGDON   CLIFFORD  85 

to  do  that.  But  Sir  William  Rowan  Hamilton,  knowing  these 
facts  and  also  the  explanation  of  them  which  Fresnel  had  given, 
thought  about  the  subject,  and  he  predicted  that  by  looking 
through  certain  crystals  in  a  particular  direction  we  should  see 
not  two  dots  but  a  continuous  circle.  Mr.  Lloyd  made  the 
experiment,  and  saw  the  circle,  a  result  which  had  never  been 
even  suspected.  This  has  always  been  considered  one  of  the 
most  signal  instances  of  scientific  thought  in  the  domain  of 
physics.  It  is  most  distinctly  an  application  of  experience 
gained  under  certain  circumstances  to  entirely  different  cir- 
circumstances." 

In  physical  science  there  are  two  kinds  of  law — distinguished 
as  "empirical"  and  "rational."  The  former  expresses  a  relation 
which  is  sufficiently  true  for  practical  purposes  and  within 
certain  limits;  for  example,  many  of  the  formulas  used  by  engi- 
neers. But  a  rational  law  states  a  connection  which  is  accu- 
rately true,  without  any  modification  of  limit.  In  the  theorems 
of  geometry  we  have  examples  of  scientific  exactness;  for 
example,  in  the  theorem  that  the  sum  of  the  three  interior 
angles  of  a  plane  triangle  is  equal  to  two  right  angles.  The 
equality  is  one  not  of  approximation,  but  of  exactness.  Now 
the  philosopher  Kant  pointed  to  such  a  truth  and  said :  We  know 
that  it  is  true  not  merely  here  and  now,  but  everywhere  and  for 
all  time;  such  knowledge  cannot  be  gained  by  experience ;  there 
must  be  some  other  source  of  such  knowledge.  His  solution 
was  that  space  and  time  are  forms  of  the  sensibility;  that  truths 
about  them  are  not  obtained  by  empirical  induction,  but  by 
means  of  intuition;  and  that  the  characters  of  necessity  and 
universality  distinguished  these  truths  from  other  truths.  This 
philosophy  was  accepted  by  Sir  William  Rowan  Hamilton,  and 
to  him  it  was  not  a  barren  philosophy,  for  it  served  as  the 
starting  point  of  his  discoveries  in  algebra  which  culminated  in 
the  discovery  of  quaternions. 

This  philosophy  was  admired  but  not  accepted  by  Clifford; 
he  was,  so  long  as  he  lived,  too  strongly  influenced  by  the 
philosophy  which  has  been  built  upon  the  theory  of  evolution. 
He  admits  that  the  only  way  of  escape  from  Kant's  conclusions 


86  TEN  BRITISH  MATHEMATICIANS 

is  by  denying  the  theoretical  exactness  of  the  proposition  referred 
to.  He  says,  "About  the  beginning  of  the  present  century  the 
foundations  of  geometry  were  criticised  independently  by  two 
mathematicians,  Lobatchewsky  and  Gauss,  whose  results  have 
been  extended  and  generalized  more  recently  by  Riemann  and 
Helmholtz.  And  the  conclusion  to  which  these  investigations 
lead  is  that,  although  the  assumptions  which  were  very  properly 
made  by  the  ancient  geometers  are  practically  exact — that  is 
to  say,  more  exact  than  experiment  can  be — for  such  finite  things 
as  we  have  to  deal  with,  and  such  portions  of  space  as  we  can 
reach;  yet  the  truth  of  them  for  very  much  larger  things,  or 
very  much  smaller  things,  or  parts  of  space  which  are  at  present 
beyond  our  reach,  is  a  matter  to  be  decided  by  experiment, 
when  its  powers  are  considerably  increased.  I  want  to  make 
as  clear  as  possible  the  real  state  of  this  question  at  present, 
because  it  is  often  supposed  to  be  a  question  of  words  or  meta- 
physics, whereas  it  is  a  very  distinct  and  simple  question  of 
fact.  I  am  supposed  to  know  that  the  three  angles  of  a  recti- 
linear triangle  are  exactly  equal  to  two  right  angles.  Now 
suppose  that  three  points  are  taken  in  space,  distant  from  one 
another  as  far  as  the  Sun  is  from  a  Centauri,  and  that  the 
shortest  distances  between  these  points  are  drawn  so  as  to  form 
a  triangle.  And  suppose  the  angles  of  this  triangle  to  be  very 
accurately  measured  and  added  together;  this  can  at  present 
be  done  so  accurately  that  the  error  shall  certainly  be  less  than 
one  minute,  less  therefore  than  the  five-thousandth  part  of  a 
right  angle.  Then  I  do  not  know  that  this  sum  would  differ 
at  all  from  two  right  angles;  but  also  I  do  not  know  that  the 
difference  would  be  less  than  ten  degrees  or  the  ninth  part  of 
a  right  angle." 

You  will  observe  that  Clifford's  philosophy  depends  on  the 
validity  of  Lobatchewsky's  ideas.  Now  it  has  been  shown  by 
an  Italian  mathematician,  named  Beltrami,  that  the  plane 
geometry  of  Lobatchewsky  corresponds  to  trigonometry  on  a 
surface  called  the  pseudosphere.  Clifford  and  other  followers  of 
Lobatchewsky  admit  Beltrami's  interpretation,  an  interpretation 
which  does  not  involve  any  paradox  about  geometrical  space, 


WILLIAM   KINGDOM   CLIFFORD  87 

and  which  leaves  the  trigonometry  of  the  plane  alone  as  a  dif- 
ferent thing.  If  that  interpretation  is  true,  the  Lobatchewskian 
plane  triangle  is  after  all  a  triangle  on  a  special  surface,  and  the 
straight  lines  joining  the  points  are  not  the  shortest  absolutely, 
but  only  the  shortest  with  respect  to  the  surface,  whatever  that 
may  mean.  If  so,  then  Clifford's  argument  for  the  empirical 
nature  of  the  proposition  referred  to  fails;  and  nothing  pre- 
vents us  from  falling  back  on  Kant's  position,  namely,  that 
there  is  a  body  of  knowledge  characterized  by  absolute  exact- 
ness and  possessing  universal  application  in  time  and  space; 
and  as  a  particular  case  thereof  we  believe  that  the  sum  of  the 
three  angles  of  Clifford's  gigantic  triangle  is  precisely  two  right 
angles. 

Trigonometry  on  a  spherical  surface  is  a  generalized  form  of 
plane  trigonometry,  from  the  theorems  of  the  former  we  can 
deduce  the  theorems  of  the  latter  by  supposing  the  radius  of 
the  sphere  to  be  infinite.  The  sum  of  the  three  angles  of  a 
spherical  triangle  is  greater  than  two  right  angles;  the  sum  of 
the  angles  of  a  plain  triangle  is  equal  to  two  right  angles;  we 
infer  that  there  is  another  surface,  complementary  to  the 
sphere,  such  that  the  angles  of  any  triangle  on  it  are  less  than 
two  right  angles.  The  complementary  surface  to  which  I  refer 
is  not  the  pseudosphere,  but  the  equilateral  hyperboloid.  As 
the  plane  is  the  transition  surface  between  the  sphere  and 
the  equilateral  hyperboloid,  and  a  triangle  on  it  is  the  transi- 
tion triangle  between  the  spherical  triangle  and  the  equilateral 
hyperboloidal  triangle,  the  sum  of  the  angles  of  the  plane  tri- 
angle must  be  exactly  equal  to  two  right  angles. 

In  1873,  the  British  Association  met  at  Bradford;  on  this 
occasion  the  evening  discourse  was  delivered  by  Maxwell, 
the  celebrated  physicist.  He  chose  for  his  subject  "  Mole- 
cules." The  application  of  the  method  of  spectrum-ana  ysis 
assures  the  physicist  that  he  can  find  out  in  his  laboratory 
truths  of  universal  validity  in  space  and  t  me.  In  fact,  the 
chief  maxim  of  physical  science,  according  to  Maxwell  is, 
that  physical  changes  are  independent  of  the  conditions  of 
space  and  time,  and  depend  only  on  conditions  of  configuration 


88  TEN   BRITISH   MATHEMATICIANS 

of  bodies,  temperature,  pressure,  etc.  The  address  closed 
with  a  celebrated  passage  in  striking  contrast  to  Clifford's 
address:  "  In  the  heavens  we  discover  by  their  light,  and 
by  their  light  alone,  stars  so  distant  from  each  other  that  no 
material  thing  can  ever  have  passed  from  one  to  another;  and 
yet  this  light,  which  is  to  us  the  sole  evidence  of  the  exis  ence 
of  these  distant  worlds,  tells  us  also  that  each  of  th  m  is  built 
up  of  molecules  of  the  same  kinds  as  those  which  are  found  on 
earth.  A  mol  cule  of  hydrogen,  for  example,  whether  in  S  rius 
or  in  Arcturus,  executes  its  vibrations  in  precisely  the  same 
time.  No  theory  of  evolution  can  be  formed  to  account  for 
the  similarity  of  molecules,  for  evolution  necessarily  implies 
continuous  change,  and  the  molecule  is  incapable  of  growth 
or  decay,  of  generation  or  destruction.  None  of  the  processes 
of  Nature  since  the  time  when  Nature  began,  have  produced 
the  slightest  difference  in  the  properties  of  any  molecule.  We 
are  therefore  unable  to  ascribe  either  the  existence  of  the  mole- 
cules or  the  identity  of  their  properties  to  any  of  the  causes 
which  we  call  natural.  On  the  other  hand,  the  exact  equality 
of  each  molecule  to  all  others  of  the  same  kind  gives  it,  as 
Sir  John  Herschel  has  well  said,  the  essential  character  of  a 
manufactured  article,  and  precludes  the  idea  of  its  being  eternal 
and  self -existent." 

What  reply  could  Clifford  make  to  this?  In  a  discourse 
on  the  "  First  and  last  catastrophe  "  delivered  soon  afterwards, 
he  said  "  If  anyone  not  possessing  the  great  authority  of 
Maxwell,  had  put  forward  an  argument,  founded  upon  a  sci- 
entific basis,  in  which  there  occurred  assumptions  about  what 
things  can  and  what  things  cannot  have  existed  from  eternity, 
and  about  the  exact  similarity  of  two  or  more  things  established 
by  experiment,  we  would  say:  '  Past  eternity;  absolute  exact- 
ness; won't  do  ';  and  we  should  pass  on  to  another  book. 
The  experience  of  all  scientific  culture  for  all  ages  during  which 
it  has  been  a  light  to  men  has  shown  us  that  we  never  do  get 
at  any  conclusions  of  that  sort.  We  do  not  get  at  conclusions 
about  infinite  time,  or  .nfinite  exactness.  We  get  at  conclu- 
sions which  are  as  nearly  true  as  exper'ment  can  show,  and 


WILLIAM   KINGDON      CLIFFORD  89 

sometimes  which  are  a  great  deal  more  correct  than  direct 
experiment  can  be,  so  that  we  are  able  actually  to  correct 
one  experiment  by  deductions  from  another,  but  we  never 
get  at  conclusions  which  we  have  a  right  to  say  are  absolutely 
exact." 

Clifford  had  not  faith  in  the  exactness  of  mathematical  science 
nor  faith  in  that  maxim  of  physical  science  which  has  built 
up  the  new  astronomy,  and  extended  all  the  bounds  of  physical 
science.  Faith  in  an  exact  order  of  Nature  was  the  charac- 
teristic of  Faraday,  and  he  was  by  unanimous  consent  the 
greatest  electrician  of  the  nineteenth  century.  What  is  the 
general  direction  of  progress  in  science?  Physics  is  becoming 
more  and  more  mathematical;  chemistry  is  becoming  more 
and  more  physical,  and  I  daresay  the  biological  sciences  are 
moving  in  the  same  direction.  They  are  all  moving  towards 
exactness;  consequently  a  true  philosophy  of  science  will  de- 
pend on  the  principles  of  mathematics  much  more  than 
upon  the  phenomena  of  biology.  Clifford,  I  believe,  had  he 
lived  longer,  would  have  changed  his  philosophy  for  a  more 
mathematical  one.  In  1874  there  appeared  in  Nature  among 
the  letters  from  correspondents  one  to  the  following  effect: 

An  anagram  :  The  practice  of  enclosing  discoveries  in  sealed 
packets  and  sending  them  to  Academies  seems  so  inferior  to 
the  old  one  of  Huyghens,  that  the  following  is  sent  you  for 
publication  in  the  old  conservated  form: 


This  anagram  was  explained  in  a  book  entitled  The  Unseen 
Universe,  which  was  published  anonymously  in  1875;  and  is 
there  translated,  "  Thought  conceived  to  affect  the  matter  of 
another  universe  simultaneously  with  this  may  explain  a  future 
state."  The  book  was  evidently  a  work  of  a  physicist  or 
physicists,  and  as  phys:cists  were  not  so  numerous  then  as 
they  are  now,  it  was  not  difficult  to  determine  the  authorship 
from  internal  evidence.  It  was  attributed  to  Tait,  the  professor 
of  physics  at  Edinburgh  University,  and  Balfour  Stewart,  the 
professor  of  physics  at  Owens  College,  Manchester.  When 


90  TEN  BRITISH  MATHEMATICIANS 

the  fourth  edition  appeared,  their  names  were  given  on  the 
title  page. 

The  kernel  of  the  book  is  the  above  so-called  discovery, 
first  published  in  the  form  of  an  anagram.  Preliminary  chap- 
ters are  devoted  to  a  survey  of  the  beliefs  of  ancient  peoples 
on  the  subject  of  the  immortality  of  the  soul;  to  physical 
axioms;  to  the  physical  doctrine  of  energy,  matter,  and  ether; 
and  to  the  biological  doctrine  of  development;  in  the  last 
chapter  we  come  to  the  unseen  universe.  What  is  meant  by 
the  unseen  universe?  Matter  is  made  up  of  molecules,  which 
are  supposed  to  be  vortex-rings  of  an  imperfect  fluid,  namely, 
the  luminiferous  ether;  the  luminous  ether  is  made  up  of  much 
smaller  molecules,  which  are  vortex-rings  in  a  second  ether. 
These  smaller  molecules  with  the  ether  in  which  they  float 
are  the  unseen  universe.  The  authors  see  reason  to  believe 
that  the  unseen  universe  absorbs  energy  from  the  visible  uni- 
verse and  vice  versa.  The  soul  is  a  frame  which  is  made  of 
the  refined  molecules  and  exists  in  the  unseen  universe.  In 
life  it  is  attached  to  the  body.  Every  thought  we  think  is 
accompanied  by  certain  motions  of  the  coarse  molecules  of 
the  brain,  these  motions  are  propagated  through  the  visible 
universe,  but  a  part  of  each  motion  is  absorbed  by  the  fine 
molecules  of  the  soul.  Consequently  the  soul  has  an  organ 
of  memory  as  well  as  the  body;  at  death  the  soul  with  its 
organ  of  memory  is  simply  set  free  from  associat'on  with  the 
coarse  molecules  of  the  body.  In  this  way  the  authors  con- 
sider that  they  have  shown  the  physical  possibility  of  the 
immortality  of  the  soul. 

The  curious  part  of  the  book  follows:  the  authors  change 
their  possibility  into  a  theory  and  apply  it  to  explain  the  main 
doctrines  of  Christianity;  and  it  is  certainly  remarkable  to  find 
in  the  same  book  a  discussion  of  Carnot's  heat-engine  and  ex- 
tensive quotations  from  the  apostles  and  prophets.  Clifford 
wrote  an  elaborate  review  which  he  finished  in  one  sitting  occu- 
pying twelve  hours.  He  pointed  out  the  difficulties  to  which 
the  main  speculation,  which  he  admitted  to  be  ingenious,  is 
liable;  but  his  wrath  knew  no  bounds  when  he  proceeded  to  con- 


WILLIAM  KINGDON   CLIFFORD  91 

sider  the  application  to  the  doctrines  of  Christianity;  for  from 
being  a  High  Churchman  in  youth  he  became  an  agnostic  in 
later  years ;  and  he  could  not  write  on  any  religious  question 
without  using  language  which  was  offensive  even  to  his  friends. 
The  Phaedo  of  Plato  is  more  satisfying  to  the  mind  than  the 
Unseen  Universe  of  Tait  and  Stewart.  In  it,  Socrates  discusses 
with  his  friends  the  immortality  of  the  soul,  just  before  taking 
the  draught  of  poison.  One  argument  he  advances  is,  How  can 
the  works  of  an  artist  be  more  enduring  than  the  artist  himself? 
This  is  a  question  which  comes  home  in  force  when  we  peruse 
the  works  of  Peacock,  De  Morgan,  Hamilton,  Boole,  Cay  ley 
and  Clifford. 


HENRY  JOHN   STEPHEN   SMITH* 

(1826-1883) 

HENRY  JOHN  STEPHEN  SMITH  was  born  in  Dublin,  Ireland, 
on  November  2,  1826.  His  father,  John  Smith,  was  an  Irish 
barrister,  who  had  graduated  at  Trinity  College,  Dublin,  and 
had  afterwards  studied  at  the  Temple,  London,  as  a  pupil  of 
Henry  John  Stephen,  the  editor  of  Blackstone's  Commentaries; 
hence  the  given  name  of  the  future  mathematician.  His  mother 
was  Ma-y  Murphy,  an  accomplished  and  clever  Irishwoman, 
tall  and  beautiful.  Henry  was  the  youngest  of  four  children, 
and  was  but  two  years  old  when  his  father  died.  His  mother 
would  have  been  left  in  straitened  circumstances  had  she  not 
been  successful  in  claiming  a  bequest  of  £10,000  which  had  been 
left  to  her  husband  but  had  been  disputed.  On  receiving  this 
money,  she  migrated  to  England,  and  finally  settled  in  the  Isle 
of  Wight. 

Henry  as  a  child  was  sickly  and  very  near-sighted.  When 
four  years  of  age  he  displayed  a  genius  for  mastering  languages. 
His  first  instructor  was  his  mother,  who  had  an  accurate  knowl- 
edge of  the  class'cs.  When  eleven  years  of  age,  he,  along  with 
his  brother  and  sisters,  was  placed  in  the  charge  of  a  private 
tutor,  who  was  strong  in  the  classics;  in  one  year  he  read  a 
large  portion  of  the  Greek  and  Latin  authors  commonly  studied. 
His  tutor  was  'impressed  with  his  power  of  memory,  quickness 
of  perception,  indefatigable  diligence,  and  intuitive  grasp  of 
whatever  he  studied.  In  their  leisure  hours  the  children  would 
improvise'  plays  from  Homer,  or  Robinson  Crusoe;  and  they 
also  became  diligent  students  of  animal  and  insect  life.  Next 
year  a  new  tutor  was  strong  in  the  mathematics,  and  with  his 
aid  Henry  became  acquainted  with  advanced  arithmetic,  and 

*  This  Lecture  was  Delivered  March  15,  1902. — EDITORS. 
92 


HENRY  JOHN   STEPHEN   SMITH  93 

the  elements  of  algebra  and  geometry.  The  year  following, 
Mrs.  Smith  moved  to  Oxford,  and  placed  Henry  under  the 
care  of  Rev.  Mr.  Highton,  who  was  not  only  a  sound  scholar, 
but  an  exceptionally  good  mathematician.  The  year  following 
Mr.  Highton  received  a  mastership  at  Rugby  with  a  boarding- 
house  attached  to  it  (wnich  is  important  from  a  financial  point 
of  view)  and  he  took  Henry  Smith  with  him  as  his  first  boarder. 
Thus  at  the  age  of  fifteen  Henry  Smith  was  launched  into  the 
life  of  the  English  public  school,  and  Rugby  was  then  under  the 
most  famous  headmaster  of  the  day,  Dr.  Arnold.  Schoolboy  life 
as  it  was  then  at  Rugby  has  been  depicted  by  Hughes  in  "  Tom 
Brown's  Schooldays." 

Here  he  showed  great  and  all-around  ability.  It  became 
his  ambition  to  crown  his  school  career  by  carrying  off  an 
entrance  scholarship  at  Balliol  College,  Oxford.  But  as  a  sister 
and  brother  had  already  died  of  consumption,  his  mother  did 
not  allow  him  to  complete  his  third  and  final  year  at  Rugby,  but 
took  him  to  Italy,  where  he  continued  his  reading  privately. 
Notwithstanding  this  manifest  disadvantage,  he  was  able  to 
carry  off  the  coveted  scholarship;  and  at  the  age  of  nineteen 
he  began  residence  as  a  student  of  Balliol  College.  The  next 
long  vacation  was  spent  in  Italy,  and  there  his  health  broke 
down.  By  the  following  winter  he  had  not  recovered  enough 
to  warrant  his  return  to  Oxford;  instead,  he  went  to  Paris, 
and  took  several  of  the  courses  at  the  Sorbonne  and  the  College 
de  France.  These  studies  abroad  had  much  influence  on  his 
future  career  as  a  mathematician.  Thereafter  he  resumed  his 
undergraduate  studies  at  Oxford,  carried  off  what  is  considered 
the  highest  classical  honor,  and  in  1849,  when  23  years  old, 
finished  his  undergraduate  career  with  a  double-first;  that  is, 
in  the  honors  examination  for  bache  or  of  arts  he  took  first-class 
rank  in  the  classics,  and  also  first-class  rank  in  he  math  matics. 

It  is  not  very  pleasant  to  be  a  double  fi  st,  for  the  outwardly 
envied  and  distinguished  recipient  is  apt  to  find  himself  in  the 
position  of  the  ass  between  two  equally  inviting  bundles  of  hay, 
unless  indeed  there  is  some  external  attraction  superior  to  both. 
In  the  case  of  Smith,  the  external  attraction  was  the  bar,  for 


94  TEN   BRITISH   MATHEMATICIANS 

which  he  was  in  many  respects  well  suited;  but  the  feebleness 
of  his  constitution  led  him  to  abandon  that  course.  So  he  had 
a  difficulty  in  deciding  between  classics  and  mathematics,  and 
there  is  .a  story  to  the  effect  that  he  finally  solved  the  difficulty 
by  tossing  up  a  penny.  He  certainly  used  the  expression:  but 
the  reasons  which  determined  his  choice  in  favor  of  mathematics 
were  first,  his  weak  sight,  which  made  thinking  preferable  to 
reading,  and  secondly,  the  opportunity  which  presented  itself. 

At  that  time  Oxford  was  recovering  from  the  excitement 
which  had  been  produced  by  the  Tractarian  movement,  and 
which  had  ended  in  Newman  going  over  to  the  Church  of  Rome. 
But  a  Parliamentary  Commission  had  been  appointed  to  inquire 
into  the  working  of  the  University.  The  old  system  of  close 
scholarships  and  fellowships  was  doomed,  and  the  close  pre- 
serves of  the  Colleges  were  being  either  extinguished  or  thrown 
open  to  public  competition.  Resident  professors,  married  tutors 
or  fellows  were  almost  or  quite  unknown;  the  heads  of  the 
several  colleges,  then  the  governing  body  of  the  University, 
formed  a  little  society  by  themselves.  Balliol  College  (founded 
by  John  Balliol,  the  unfortunate  King  of  Scotland  who  was 
willing  to  sell  its  independence)  was  then  the  most  distinguished 
for  intellectual  eminence;  the  master  was  singular  among  his 
compeers  for  keeping  steadily  in  view  the  true  aim  of  a  col- 
lege, and  he  reformed  the  abuses  of  privilege  and  close  endow- 
ment as  far  as  he  legally  could.  Smith  was  elected  a  fellow 
with  the  hope  that  he  would  consent  to  reside,  and  take  the 
further  office  of  tutor  in  mathematics,  which  he  did.  Soon 
after  he  became  one  of  the  mathematical  tutors  of  Balliol  he 
was  asked  by  his  college  to  deliver  a  course  of  lectures  on 
chemistry.  For  this  purpose  he  took  up  the  study  of  chemical 
analysis,  and  exhibited  skill  in  manipulation  and  accuracy  in 
work.  He  had  an  idea  of  seeking  numerical  relations  connecting 
the  atomic  weights  of  the  elements,  and  some  mathematical 
basis  for  their  properties  which  might  enable  experiments  to 
be  predicted  by  the  operation  of  the  mind. 

About  this  time  Whewell,  the  master  of  Trinity  College, 
Cambridge,  wrote  The  Plurality  of  Worlds,  which  was  at  first 


HENRY   JOHN   STEPHEN   SMITH  95 

published  anonymously.  Whewell  pointed  out  what  he  called 
law  of  waste  traceable  in  the  Divine  economy;  and  his  argu- 
ment was  that  the  other  planets  were  waste  effects,  the  Earth 
the  only  oasis  in  the  desert  of  our  system,  the  only  world 
inhabited  by  intelligent  beings;  Sir  David  Brewster,  a  Scottish 
physicist,  inventor  of  the  kaleidoscope,  wrote  a  fiery  answer 
entitled  "  More  worlds  than  one,  the  creed  of  the  philosopher 
and  the  hope  of  the  Christian."  In  1855  Smith  wrote  an  essay 
on  this  subject  for  a  volume  of  Oxford  and  Cambridge  Essays 
in  which  the  fallibility  both  of  men  of  science  and  of  theologians 
was  impartially  exposed.  It  was  his  first  and  only  effort  at 
popular  writing. 

His  two  earliest  mathematical  papers  were  on  geometrical 
subjects,  but  the  third  concerned  that  branch  of  mathematics 
in  which  he  won  fame — the  theory  of  numbers.  How  he  was 
led  to  take  up  this  branch  of  mathematics  is  not  stated  on 
authority,  but  it  was  probably  as  follows:  There  was  then  no 
school  of  mathematics  at  Oxford;  the  symbolical  school  was 
flourishing  at  Cambridge;  and  Hamilton  was  lecturing  on 
Quaternions  at  Dublin.  Smith  did  not  estimate  either  of  these 
very  highly;  he  had  studied  at  Paris  under  some  of  the  great 
French  analysts;  he  had  lived  much  on  the  Continent,  and  was 
familiar  with  the  French,  German  and  Italian  languages.  As 
a  scholar  he  was  drawn  to  the  masterly  disquisitions  of  Gauss, 
who  had  made  the  theory  of  numbers  a  principal  subject  of 
research.  I  may  quote  here  his  estimate  of  Gauss  and  of  his 
work:  "  If  we  except  the  great  name  of  Newton  (and  the  excep- 
tion is  one  which  Gauss  himself  would  have  been  delighted  to 
make)  it  is  probable  that  no  mathematician  of  any  age  or  country 
has  ever  surpassed  Gauss  in  the  combination  of  an  abundant 
fertility  of  invention  with  an  absolute  vigorousness  in  demonstra- 
tion, which  the  ancient  Greeks  themselves  might  have  envied. 
It  may  be  admitted,  without  any  disparagement  to  the  eminence 
of  such  great  mathematicians  as  Euler  and  Cauchy  that  they 
were  so  overwhelmed  with  the  exuberant  wealth  of  their  own 
creations,  and  so  fascinated  >by  the  interest  attaching  to  the 
results  at  which  they  arrived,  that  they  did  not  greatly  care 


96  TEN  BRITISH  MATHEMATICIANS 

to  expend  their  time  in  arranging  their  ideas  in  a  strictly  logical 
order,  or  even  in  establishing  by  irrefragable  proof  propositions 
which  they  instinctively  felt,  and  could  almost  see  to  be  true. 
With  Gauss  the  case  was  otherwise.  It  may  seem  paradoxical, 
but  it  is  probably  nevertheless  true  that  it  is  precisely  the  effort 
after  a  logical  perfection  of  form  which  has  rendered  the  writings 
of  Gauss  open  to  the  charge  of  obscurity  and  unnecessary  diffi- 
culty. The  fact  is  that  there  is  neither  obscurity  nor  difficulty 
in  his  writings,  as  long  as  we  read  them  in  the  submissive  spirit 
in  which  an  intelligent  schoolboy  is  made  to  read  his  Euclid. 
Every  assertion  that  is  made  is  fully  proved,  and  the  assertions 
succeed  one  another  in  a  perfectly  just  analogical  order;  there 
nothing  so  far  of  which  we  can  complain.  But  when  we  have 
finished  the  perusal,  we  soon  begin  to  feel  that  our  work  is  but 
begun,  that  we  are  still  standing  on  the  threshold  of  the  temple, 
and  that  there  is  a  secret  which  lies  behind  the  veil  and  is  as 
yet  concealed  from  us.  No  vestige  appears  of  the  process  by 
which  the  result  itself  was  obtained,  perhaps  not  even  a  trace 
of  the  considerations  which  suggested  the  successive  steps  of 
the  demonstration.  Gauss  says  more  than  once  that  for  brevity, 
he  gives  only  the  synthesis,  and  suppresses  the  analysis  of  his 
propositions.  Pauca  sed  matura — few  but  well-matured — were 
the  words  with  which  he  delighted  to  describe  the  character  which 
he  endeavored  to  impress  upon  his  mathematical  writings. 
If,  on  the  other  hand,  we  turn  to  a  memoir  of  Euler's,  there 
is  a  sort  of  free  and  luxuriant  gracefulness  about  the  whole 
performance,  which  tells  of  the  quiet  pleasure  which  Euler  must 
have  taken  in  each  step  of  his  work;  but  we  are  conscious  never- 
theless that  we  are  at  an  immense  distance  from  the  severe 
grandeur  of  design  which  is  characteristic  of  all  Gauss's  greater 
efforts." 

Following  the  example  of  Gauss,  he  wrote  his  first  paper 
on  the  theory  of  numbers  in  Latin:  "  De  compositione  nume- 
rorum  primorum  formae  4^+1  ex  duobus  quadra tis."  In  it 
he  proves  in  an  original  manner  the  theorem  of  Fermat — "  That 
every  prime  number  of  the  form  4^+1  (n  being  an  integer 
number)  is  the  sum  of " two  square  numbers."  In  his  second 


HENRY  JOHN   STEPHEN   SMITH  97 

paper  he  gives  an  introduction  to  the  theory  of  numbers.  "  It 
is  probable  that  the  Pythagorean  school  was  acquainted  with 
the  definition  and  nature  of  prime  numbers;  nevertheless  the 
arithmetical  books  of  the  elements  of  Euclid  contain  the  oldest 
extant  investigations  respecting  them;  and,  in  particular  the 
celebrated  yet  simple  demonstration  that  the  number  of  the 
primes  is  infinite.  To  Eratosthenes  of  Alexandria,  who  is  for 
so  many  other  reasons  entitled  to  a  place  in  the  history  of  the 
sciences,  is  attributed  the  invention  of  the  method  by  which  the 
primes  may  successively  be  determined  in  order  of  magnitude. 
It  is  termed,  after  him,  '  the  sieve  of  Eratosthenes  ';  and  is  es- 
sentially a  method  of  exclusion,  by  which  all  composite  numbers 
are  successively  erased  from  the  series  of  natural  numbers,  and 
the  primes  alone  are  left  remaining.  It  requires  only  one  kind 
of  arithmetical  operation;  that  is  to  say,  the  formation  of  the 
successive  multiples  of  given  numbers,  or  in  other  words, 
addition  only.  Indeed  it  may  be  said  to  require  no  arithmetical 
operation  whatever,  for  if  the  natural  series  of  numbers  be 
represented  by  points  set  off  at  equal  distances  along  a  line, 
by  using  a  geometrical  compass  we  can  determine  without  cal- 
culation the  multiples  of  any  given  number.  And  in  fact,  it 
was  by  a  mechanical  contrivance  of  this  nature  that  M.  Burck- 
hardt  calculated  his  table  of  the  least  divisors  of  the  first  three 
millions  of  numbers. 

In  1857  Mrs.  Smith  died;  as  the  result  of  her  cares  and 
exertions  she  had  seen  her  son  enter  Balliol  College  as  a 
scholar,  graduate  a  double-first,  elected  a  fellow  of  his  college, 
appointed  tutor  in  mathematics,  and  enter  on  his  career  as  an 
independent  mathematician.  The  brother  and  sister  that  were 
left  arranged  to  keep  house  in  Oxford,  the  two  spending  the 
terms  together,  and  each  being  allowed  complete  liberty  of  move- 
ment during  the  vacations.  Thereafter  this  was  the  domestic 
arrangement  in  which  Smith  lived  and  worked ;  he  never  married. 
As  the  owner  of  a  house,  instead  of  living  in  rooms  in  college 
he  was  able  to  satisfy  his  fondness  for  pet  animals,  and  also 
to  extend  Irish  hospitality  to  visiting  friends  under  his  own 
roof.  He  had  no  household  cares  to  destroy  the  needed  serenity 


98  TEN   BRITISH   MATHEMATICIANS 

for  scientific  work,  excepting  that  he  was  "careless  in  money 
matters,  and  trusted  more  to  speculation  in  mining  shares  than 
to  economic  management  of  his  income.  Though  addicted  to 
the  theory  of  numbers,  he  was  not  in  any  sense  a  recluse;  on 
the  contrary  he  entered  with  zest  into  every  form  of  social 
enjoyment  in  Oxford,  from  croquet  parties  and  picnics  to  ban- 
quets. He  had  the  rare  power  of  utilizing  stray  hours  of  leisure, 
and  it  was  in  such  odd  times  that  he  accomplished  most  of  his 
scientific  work.  After  attending  a  picnic  in  the  afternoon,  he 
could  mount  to  those  serene  heights  in  the  theory  of  numbers 

"Where  never  creeps  a  cloud  or  moves  a  wind, 
Nor  ever  falls  the  least  white  star  of  snow, 
Nor  ever  lowest  roll  of  thunder  moans, 
Nor  sound  of  human  sorrow  mounts,  to  mar 
Their  sacred  everlasting  calm." 

Then  he  could  of  a  sudden  come  down  from  these  heights 
to  attend  a  dinner,  and  could  conduct  himself  there,  not  as  a 
mathematical  genius  lost  in  reverie  and  pointed  out  as  a  poor 
and  eccentric  mortal,  but  on  the  contrary  as  a  thorough  man 
of  the  world  greatly  liked  by  everybody. 

In  1860,  when  Smith  was  34  years  old,  the  Savilian  professor 
of  geometry  at  Oxford  died.  At  that  time  the  English  uni- 
versities were  so  constituted  that  the  teaching  was  done  by  the 
college  tutors.  The  professors  were  officers  of  the  University; 
and  before  reform  set  in,  they  not  only  did  not  teach,  they  did 
not  even  reside  in  Oxford.  At  the  present  day  the  lectures  of 
the  University  professors  are  in  general  attended  by  only  a  few 
advanced  students.  Henry  Smith  was  the  only  Oxford  candi- 
date; there  were  other  candidates  from  the  outside,  among 
them  George  Boole,  then  professor  of  mathematics  at  Queens 
College,  Cork.  Smith's  claims  and  talents  were  considered  so 
conspicuous  by  the  electors,  that  they  did  not  consider  any 
other  candidates.  He  did  not  resign  as  tutor  at  Balliol,  but 
continued  to  discharge  the  arduous  duties,  in  order  that  the 
income  of  his  Fellowship  might  be  continued.  With  proper 
financial  sense  he  might  have  been  spared  from  labors  which 
militated  against  the  discharge  of  the  higher  duties  of  professor. 


HENRY  JOHN  STEPHEN  SMITH  99 

His  freedom  during  vacation  gave  him  the  opportunity  of 
attending  the  meetings  of  the  British  Association,  where  he 
was  not  only  a  distinguished  savant,  but  an  accomplished 
member  of  the  social  organization  known  as  the  Red  Lions.  In 
1858  he  was  selected  by  that  body  to  prepare  a  report  upon  the 
Theory  of  Numbers.  It  was  prepared  in  five  parts,  extending 
over  the  years  1859-1865.  It  is  neither  a  history  nor  a  treatise, 
but  something  intermediate.  The  author  analyzes  with  remark- 
able clearness  and  order  the  works  of  mathematicians  for  the 
preceding  century  upon  the  theory  of  congruences,  and  upon 
that  of  binary  quadratic  forms.  He  returns  to  the  original 
sources,  indicates  the  principle  and  sketches  the  course  of  the 
demonstrations,  and  states  the  result,  often  adding  something 
of  his  own.  The  work  has  been  pronounced  to  be  the  most 
complete  and  elegant  monument  ever  erected  to  the  theory  of 
numbers,  and  the  model  of  what  a  scientific  report  ought  to  be. 

During  the  preparation  of  the  Report,  and  as  a  logical  con- 
sequence of  the  researches  connected  therewith,  Smith  pub- 
lished several  original  contributions  to  the  higher  rithmetic. 
Some  were  in  complete  form  and  appeared  in  the  Philosophical 
Transactions  of  the  Royal  Society  of  London;  others  were 
incomplete,  giving  only  the  results  without  the  extended  demon- 
strations, and  appeared  in  the  Proceedings  of  that  Society.  One 
of  the  latter,  entitled  "  On  the  orders  and  genera  of  quadratic 
forms  containing  more  than  three  inde terminates,"  enunciates 
certain  general  principles  by  means  of  which  he  solves  a  problem 
proposed  by  Eisenstein,  namely,  the  decomposition  of  integer 
numbers  into  the  sum  of  five  squares;  and  further,  the  analogous 
problem  for  seven  squares.  It  was  also  indicated  that  the  four, 
six,  and  eight-square  theorems  of  Jacobi,  Eisenstein  and  Lion- 
ville  were  deducible  from  the  principles  set  forth. 

In  1868  he  returned  to  the  geometrical  researches  which  had 
first  occupied  his  attention.  For  a  memoir  on  "  Certain  cubic 
and  biquadratic  problems  "  the  Royal  Academy  of  Sciences  of 
Berlin  awarded  him  the  Steiner  prize.  On  account  of  his  ability 
as  a  man  of  affairs,  Smith  was  in  great  demand  for  University 
and  scientific  work  of  the  day.  He  was  made  Keeper  of  the 


100  TEN    BRITISH   MATHEMATICIANS 

University  Museum;  he  accepted  the  office  of  Mathematical 
Examiner  to  the  University  of  London;  he  was  a  member  of  a 
Royal  Commission  appointed  to  report  on  Scientific  Education; 
a  member  of  the  Commission  appointed  to  reform  the  University 
of  Oxford;  chairman  of  the  committee  of  scientists  who  were 
given  charge  of  the  Meteorological  Office,  etc.  It  was  not  till 
1873,  when  offered  a  Fellowship  by  Corpus  Christi  College, 
that  he  gave  up  his  tutorial  duties  at  Balliol.  The  demands 
of  these  offices  and  of  social  functions  upon  his  time  and  energy 
necessarily  reduced  the  total  output  of  mathematical  work  of 
the  highest  order;  the  results  of  long  research  lay  buried  in 
note-books,  and  the  necessary  time  was  not  found  for  elabora- 
ting them  into  a  form  suitable  for  publication.  Like  his  master, 
Gauss,  he  had  a  high  ideal  of  what  a  scientific  memoir  ought 
to  be  in  logical  order,  vigor  of  demonstration  and  literary  execu- 
tion; and  it  was  to  his  mathematical  friends  matter  of  regret 
that  he  did  not  reserve  more  of  his  energy  for  the  work  for 
which  he  was  exceptionally  fitted. 

He  was  a  brilliant  talker  and  wit.  Working  in  the  purely 
speculative  region  of  the  theory  of  numbers,  it  was  perhaps 
natural  that  he  should  take  an  anti-utilitarian  view  of  mathe- 
matical science,  and  that  he  should  express  it  in  exaggerated 
terms  as  a  defiance  to  the  grossly  utilitarian  views  then  pop- 
ular. It  is  reported  that  once  in  a  lecture  after  explaining 
a  new  solution  of  an  old  problem  he  said,  "  It  is  the  peculiar 
beauty  of  this  method,  gentlemen,  and  one  which  endears  it 
to  the  really  scientific  mind,  that  under  no  circumstances  can  it 
be  of  the  smallest  possible  utility."  I  believe  that  it  was  at 
a  banquet  of  the  Red  Lions  that  he  proposed  the  toast  "  Pure 
mathematics;  may  it  never  be  of  any  use  to  any  one." 

I  may  mention  some  other  specimens  of  his  wit.  "  You 
take  tea  in  the  morning,"  was  the  remark  with  which  he  once 
greeted  a  friend;  "  if  I  did  that  I  should  be  awake  all  day." 
Some  one  mentioned  to  him  the  enigmatical  motto  of  Marischal 
College,  Aberdeen:  "  They  say;  what  say  they;  let  them  say." 
"Ah,"  said  he,  "  it  expresses  the  three  stages  of  an  undergradu- 
ate's career.  '  They  say  ' — in  his  first  year  he  accepts  every- 


HENRY   JOHN   STEPHEN   SMITH  101 

thing  he  is  told  as  if  it  were  inspired.  '  What  say  they  ' — in 
his  second  year  he  is  skeptical  and  asks  that  question.  '  Let 
them  say  '  expresses  the  attitude  of  contempt  characteristic  of 
his  third  year."  Of  a  brilliant  writer  but  illogical  thinker  he 
said  "  He  is  never  right  and  never  wrong;  he  is  never  to  the 
point."  Of  Lockyer,  the  astronomer,  who  has  been  for  many 
years  the  editor  of  the  scientific  journal  Nature,  he  said, 
"  Lockyer  sometimes  forgets  that  he  is  only  the  editor,  not 
the  author,  of  Nature."  Speaking  to  a  newly  elected  fellow 
of  his  college  he  advised  him  in  a  low  whisper  to  write  a  little 
and  to  save  a  little,  adding  "  I  have  done  neither." 

At  the  jubilee  meeting  of  the  British  Association  held  at 
York  in  1881,  Prof.  Huxley  and  Sir  John  Lubbock  (now  Lord 
Avebury)  strolled  down  one  afternoon  to  the  Minster,  which 
is  considered  the  finest  cathedral  in  England.  At  the  main 
door  they  met  Prof.  Smith  coming  out,  who  made  a  mock 
movement  of  surprise.  Huxley  said,  "  You  seem  surprised  to 
see  me  here."  "  Yes,"  said  Smith,  "going  in,  you  know;  I  would 
not  have  been  surprised  to  see  you  on  one  of  the  pinnacles." 
Once  I  was  introduced  to  him  at  a  garden  party,  given  in  the 
grounds  of  York  Minster.  He  was  a  tall  man,  with  sandy  hair 
and  beard,  decidedly  good-looking,  with  a  certain  intellectual 
distinction  in  his  features  and  expression.  He  was  everywhere 
and  known  to  everyone,  the  life  and  soul  of  the  gathering.  He 
retained  to  the  day  of  his  death  the  simplicity  and  high  spirits 
of  a  boy.  Socially  he  was  an  embodiment  of  Irish  blarney 
modified  by  Oxford  dignity. 

In  1873  the  British  Association  met  at  Bradford;  at  which 
meeting  Maxwell  delivered  his  famous  "  Discourse  on  Mole- 
cules." At  the  same  meeting  Smith  was  the  president  of  the 
section  of  mathematics  and  physics.  He  did  not  take  up  any 
technical  subject  in  his  address;  but  confined  himself  to  matters 
of  interest  in  the  exact  sciences.  He  spoke  of  the  connection 
between  mathematics  and  physics,  as  evidenced  by  the  dual 
province  of  the  section.  "  So  intimate  is  the  union  between 
mathematics  and  physics  that  probably  by  far  the  larger  part 
of  the  accessions  to  our  mathematical  knowledge  have  been 


102  TEN  BRITISH   MATHEMATICIANS 

obtained  by  the  efforts  of  mathematicians  to  solve  the'problems 
set  to  them  by  experiment,  and  to  create  for  each  successive 
class  of  phenomena  a  new  calculus  or  a  new  geometry,  as  the 
case  might  be,  which  might  prove  not  wholly  inadequate  to  the 
subtlety  of  nature.  Sometimes  indeed  the  mathematician  has 
been  before  the  physicist,  and  it  has  happened  that  when  some 
great  and  new  question  has  occurred  to  the  experimenter  or  the 
observer,  he  has  found  in  the  armory  of.  the  mathematician 
the  weapons  which  he  has  needed  ready  made  to  his  hand.  But 
much  oftener  the  questions  proposed  by  the  physicist  have 
transcended  the  utmost  powers  of  the  mathematics  of  the  time, 
and  a  fresh  mathematical  creation  has  been  needed  to  supply 
the  logical  instrument  required  to  interpret  the  new  enigma." 
As  an  example  of  the  rule  he  points  out  that  the  experiments 
of  Faraday  called  forth  the  mathematical  theory  of  Maxwell; 
as  an  example  of  the  exception  that  the  work  of  Apollonius 
on  the  conic  sections  was  ready  for  Kepler  in  investigating  the 
orbits  of  the  planets. 

At  the  time  of  the  Bradford  meeting,  education  in  the  public 
schools  and  universities  of  England  was  practically  confined 
to  the  classics  and  pure  mathematics.  In  his  address  Smith 
took  up  the  importance  of  science  as  an  educational  discipline 
in  schools;  and  the  following  sentences,  falling  as  they  did  from 
a  profound  scholar,  produced  a  powerful  effect:  "  All  knowledge 
of  natural  science  that  is  imparted  to  a  boy,  is,  or  may  be,  useful 
to  him  in  the  business  of  his  after-life;  but  the  claim  of  natural 
science  to  a  place  in  education  cannot  be  rested  upon  its  useful- 
ness only.  The  great  object  of  education  is  to  expand  and  to 
train  the  mental  faculties,  and  it  is  because  we  believe  that  the 
study  of  natural  science  is  eminently  fitted  to  further  these  two 
objects  that  we  urge  its  introduction  into  school  studies.  Science 
expands  the  minds  of  the  young,  because  it  puts  before  them 
great  and  ennobling  objects  of  contemplation;  many  of  its 
truths'are  such  as  a  child  can  understand,  and  yet  such  that  while 
in  a  measure  he  understands  them,  he  is  made  to  feel  something 
of  the  greatness,  something  of  the  sublime  regularity  and  some- 
thing of  the  impenetrable  mystery,  of  the  world  in  which  he 


HENRY  JOHN  STEPHEN  SMITH  103 

is  placed.  But  science  also  trains  the  growing  faculties,  for 
science  proposes  to  itself  truth  as  its  only  object,  and  it  presents 
the  most  varied,  and  at  the  same  time  the  most  splendid 
examples  of  the  different  mental  processes  which  lead  to  the 
attainment  of  truth,  and  which  make  up  what  we  call  reasoning. 
In  science  error  is  always  possible,  often  close  at  hand;  and  the 
constant  necessity  for  being  on  our  guard  against  it  is  one  im- 
portant part  of  the  education  which  science  supplies.  But  in 
science  sophistry  is  impossible;  science  knows  no  love  of  para- 
dox; science  has  no  skill  to  make  the  worse  appear  the  better 
reason;  science  visits  with  a  not  long  deferred  exposure  all  our 
fondness  for  preconceived  opinions,  all  our  partiality  for  views 
which  we  have  ourselves  maintained;  and  thus  teaches  the  two 
best  lessons  that  can  well  be  taught — on  the  one  hand,  the  love 
of  truth;  and  on  the  other,  sobriety  and  watchfulness  in  the  use 
of  the  understanding." 

The  London  Mathematical  Society  was  founded  in  1865. 
By  going  to  the  meetings  Prof.  Smith  was  induced  to  prepare 
for  publication  a  number  of  papers  from  the  materials  of  his 
notebooks.  He  was  for  two  years  president,  and  at  the  end 
of  his  term  delivered  an  address  "  On  the  present  state  and 
prospects  of  some  branches  of  pure  mathematics."  He  began 
by  referring  to  a  charge  which  had  been  brought  against  the 
Society,  that  its  Proceedings  showed  a  partiality  in  favor  of 
one  or  two  great  branches  of  mathematical  science  to  the  com- 
parative neglect  and  possible  disparagement  of  others.  He 
replies  in  the  language  of  a  miner.  "  It  may  be  rejoined  with 
great  plausibility  that  ours  is  not  a  blamable  partiality,  but  a 
well-grounded  preference.  So  great  (we  might  contend)  have 
been  the  triumphs  achieved  in  recent  times  by  that  combination 
of  the  newer  algebra  with  the  direct  contemplation  of  space 
which  constitutes  the  modern  geometry — so  large  has  been  the 
portion  of  these  triumphs,  which  is  due  to  the  genius  of  a  few 
great  English  mathematicians;  so  vast  and  so  inviting  has  been 
the  field  thus  thrown  open  to  research,  that  we  do  well  to  press 
along  towards  a  country  which  has,  we  might  say,  been  '  pros- 
pected '  for  us,  and  in  which  we  know  beforehand  we  cannot 


104  TEN  BRITISH   MATHEMATICIANS 

fail  to  find  something  that  will  repay  our  trouble,  rather  than 
adventure  ourselves  into  regions  where,  soon  after  the  first  step, 
we  should  have  no  beaten  tracks  to  guide  us  to  the  lucky  spots, 
and  in  which  (at  the  best)  the  daily  earnings  of  the  treasure- 
seeker  are  small,  and  do  not  always  make  a  great  show,  even 
after  long  years  of  work.  Such  regions,  however,  there  are  in 
the  realm  of  pure  mathematics,  and  it  cannot  be  for  the  interest 
of  science  that  they  should  be  altogether  neglected  by  the  rising 
generation  of  English  mathematicians.  I  propose,  therefore, 
in  the  first  instance  to  direct  your  attention  to  some  few  of  these 
comparatively  neglected  spots."  Since  then  quite  a  number  of 
the  neglected  spots  pointed  out  have  been  worked. 

In  1878  Oxford  friends  urged  him  to  come  forward  as  a  candi- 
date for  the  representation  in  Parliament  of  the  University  of 
Oxford,  on  the  principle  that  a  University  constituency  ought 
to  have  for  its  representative  not  a  mere  party  politician,  but 
an  academic  man  well  acquainted  with  the  special  needs  of  the 
University.  The  main  question  before  the  electors  was  the 
approval  or  disapproval  of  the  Jingo  war  policy  of  the  Con- 
servative Government.  Henry  Smith  had  always  been  a  Liberal 
in  politics,  university  administration,  and  religion.  The  voting 
was  influenced  mainly  by  party  considerations — Beaconsfield 
or  Gladstone — with  the  result  that  Smith  was  defeated  by  more 
than  2  to  i;  but  he  had  the  satisfaction  of  knowing  that  his 
support  came  mainly  from  the  resident  and  working  members  of 
the  University.  He  did  not  expect  success  and  he  hardly  desired 
it,  but  he  did  not  shrink  when  asked  to  stand  forward  as  the 
representative  of  a  principle  in  which  he  believed.  The  election 
over,  he  devoted  himself  with  renewed  energy  to  the  publication 
of  his  mathematical  researches.  His  report  on  the  theory  of 
numbers  had  ended  in  elliptic  functions;  and  it  was  this  subject 
which  now  engaged  his  attention. 

In  February,  1882,  he  was  surprised  to  see  in  the  Comptes 
rendus  that  the  subject  proposed  by  the  Paris  Academy  of 
Science  for  the  Grand  prix  des  sciences  mathematiques  was  the 
theory  of  the  decomposition  of  integer  numbers  into  a  sum  of 
five  squares;  and  that  the  attention  of  competitors  was  directed 


HENRY  JOHN   STEPHEN   SMITH  105 

to  the  results  announced  without  demonstration  by  Eisenstein, 
whereas  nothing  was  said  about  his  papers  dealing  with 
the  same  subject  in  the  Proceedings  of  the  Royal  Society.  He 
wrote  to  M.  Hermite  calling  his  attention  to  what  he  had  pub- 
lished; in  reply  he  was  assured  that  the  members  of  the  com- 
mission did  not  know  of  the  existence  of  his  papers,  and  he 
was  advised  to  complete  his  demonstrations  and  submit  the 
memoir  according  to  the  rules  of  the  competition.  According 
to  the  rules  each  manuscript  bears  a  motto,  and  the  correspond- 
ing envelope  containing  the  name  of  the  successful  author  is 
opened.  There  were  still  three  months  before  the  closing  of  the 
concours  (i  June,  1882)  and  Smith  set  to  work,  prepared  the 
memoir  and  despatched  it  in  time. 

Meanwhile  a  political  agitation  had  grown  up  in  favor  of 
extending  the  franchise  in  the  county  constituencies.  In  the 
towns  the  mechanic  had  received  a  vote;  but  in  the  counties 
that  power  remained  with  the  squire  and  the  farmer;  poor 
Hodge,  as  he  is  called,  was  left  out.  Henry  Smith  was  not  merely 
a  Liberal;  he  felt  a  genuine  sympathy  for  the  poor  of  his  own 
land.  At  a  meeting  in  the  Oxford  Town  Hall  he  made  a  speech 
in  favor  of  the  movement,  urging  justice  to  all  classes.  From 
that  platform  he  went  home  to  die.  When  he  spoke  he  was 
suffering  from  a  cold.  The  exposure  and  excitement  were 
followed  by  congestion  of  the  liver,  to  which  he  succumbed  on 
February  9,  1883,  in  the  57th  year  of  his  age. 

Two  months  after  his  death  the  Paris  Academy  made  their 
award.  Two  of  the  three  memoirs  sent  in  were  judged  worthy 
of  the  prize.  When  the  envelopes  were  opened,  the  authors 
were  found  to  be  Prof.  Smith  and  M.  Minkowski,  a  young 
mathematician  of  Koenigsberg,  Prussia.  No  notice  was  taken 
of  Smith's  previous  publication  on  the  subject,  and  M.  Hermite 
on  being  written  to,  said  that  he  forgot  to  bring  the  matter  to 
the  notice  of  the  commission.  It  was  admitted  that  there  was 
considerable  similarity  in  the  course  of  the  investigation  in  the 
two  memoirs.  The  truth  seems  to  be  that  M.  Minkowski 
availed  himself  of  whatever  had  been  published  on  the  sub- 
ject, including  Smith's  paper,  but  to  work  up  the  memoir 


106  TEN   BRITISH  MATHEMATICIANS 

from  that  basis  cost  Smith  himself  much  intellectual  labor, 
and  must  have  cost  Minkowski  much  more.  Minkowski  is 
now  the  chief  living  authority  in  that  high  region  of  the  theory 
of  numbers.  Smith's  work  remains  the  monument  of  one  of 
the  greatest  British  mathematicians  of  the  nineteenth  century. 


JAMES  JOSEPH  SYLVESTER* 
(1814-1897) 

JAMES  JOSEPH  SYLVESTER  was  born  in  London,  on  the  3d 
of  September,  1814.  He  was  by  descent  a  Jew.  His  father 
was  Abraham  Joseph  Sylvester,  and  the  future  mathematician 
was  the  youngest  but  one  of  seven  children.  He  received  his 
elementary  education  at  two  private  schools  in  London,  and  his 
secondary  education  at  the  Royal  Institution  in  Liverpool.  At 
the  age  of  twenty  he  entered  St.  John's  College,  Cambridge; 
and  in  the  tripos  examination  he  came  out  second  wrangler. 
The  senior  wrangler  of  the  year  did  not  rise  to  any  eminence; 
the  fourth  wrangler  was  George  Green,  celebrated  for  his  con- 
tributions to  mathematical  physics;  the  fifth  wrangler  was 
Duncan  F.  Gregory,  who  subsequently  wrote  on  the  foundations 
of  algebra.  On  account  of  his  religion  Sylvester  could  not  sign 
the  thirty-nine  articles  of  the  Church  of  England ;  and  as  a  con- 
sequence he  could  neither  receive  the  degree  of  Bachelor  of 
Arts  nor  compete  for  the  Smith's  prizes,  and  as  a  further  conse- 
quence he  was  not  eligible  for  a  fellowship.  To  obtain  a  degree 
he  turned  to  the  University  of  Dublin.  After  the  theological 
tests  for  degrees  had  been  abolished  at  the  Universities  of 
Oxford  and  Cambridge  in  1872,  the  University  of  Cambridge 
granted  him  his  well-earned  degree  of  Bachelor  of  Arts  and  also 
that  of  Master  of  Arts. 

On  leaving  Cambridge  he  at  once  commenced  to  write  papers, 
and  these  were  at  first  on  applied  mathematics.  His  first  paper 
was  entitled  "  An  analytical  development  of  Fresnel's  optical 
theory  of  crystals,"  which  was  published  in  the  Philosophical 
Magazine.  Ere  long  he  was  appointed  Professor  of  Physics  in 
University  College,  London,  thus  becoming  a  colleague  of  De 

*  A  Lecture  delivered  March  21,  1902. — EDITORS. 
107 


108  TEN   BRITISH   MATHEMATICIANS 

Morgan.  At  that  time  University  College  was  almost  the  only 
institution  of  higher  education  in  England  in  which  theological 
distinctions  were  ignored.  There  was  then  no  physical  laboratory 
at  University  College,  or  indeed  at  the  University  of  Cambridge; 
which  was  fortunate  in  the  case  of  Sylvester,  for  he  would  have 
made  a  sorry  experimenter.  His  was  a  sanguine  and  fiery 
temperament,  lacking  the  patience  necessary  in  physical  manipu- 
lation. As  it  was,  even  in  these  pre-laboratory  days  he  felt 
out  of  place,  and  was  not  long  in  accepting  a  chair  of  pure 
mathematics. 

In  1841  he  became  professor  of  mathematics  at  the  Uni- 
versity of  Virginia.  In  almost  all  notices  of  his  life  nothing 
is  said  about  his  career  there;  the  truth  is  that  after  the  short 
space  of  four  years  it  came  to  a  sudden  and  rather  tragic  ter- 
mination. Among  his  students  were  two  brothers,  fully  imbued 
with  the  Southern  ideas  about  honor.  One  day  Sylvester 
criticised  the  recitation  of  the  younger  brother  in  a  wealth  of 
diction  which  offended  the  young  man's  sense  of  honor;  he 
sent  word  to  the  professor  that  he  must  apologize  or  be  chastised. 
Sylvester  did  not  apologize,  but  provided  himself  with  a  sword- 
cane;  the  young  man  provided  himself  with  a  heavy  walking- 
stick.  The  brothers  lay  in  wait  for  the  professor;  and  when  he 
came  along  the  younger  brother  demanded  an  apology,  almost 
immediately  knocked  off  Sylvester's  hat,  and  struck  him  a  blow 
on  the  bare  head  with  hi ,  heavy  stick.  Sylvester  drew  his  sword- 
cane,  and  pierced  the  young  man  just  over  the  heart;  who  fell 
back  into  his  brother's  arms,  calling  out  "  I  am  killed."  A 
spectator,  coming  up,  urged  Sylvester  away  from  the  spot. 
Without  waiting  to  pack  his  books  the  professor  left  for  New 
York,  and  took  the  earliest  possible  passage  for  England.  The 
student  was  not  seriously  hurt;  fortunately  the  point  of  the 
sword  had  struck  fair  against  a  rib. 

Sylvester,  on  his  return  to  London,  connected  himself  with 
a  firm  of  actuaries,  his  ultimate  aim  being  to  qualify  himself 
to  practice  conveyancing.  He  became  a  student  of  the  Inner 
Temple  in  1846,  and  was  called  to  the  bar  in  1850.  He  chose 
the  same  profession  as  did  Cayley;  and  in  fact  Cayley  and 


JAMES  JOSEPH   SYLVESTER  109 

Sylvester,  while  walking  the  law-courts,  discoursed  more  on 
mathematics  than  on  conveyancing.  Cayley  was  full  of  the 
theory  of  invariants;  and  it  was  by  his  discourse  that  Sylvester 
was  induced  to  take  up  the  subject.  These  two  men  were  life- 
long fri  nds;  but  it  is  safe  to  say  that  the  permanence  of  the 
friendship  was  due  to  Cayley's  kind  and  patient  disposition. 
Recognized  as  the  leading  mathematicians  of  their  day  in  Eng- 
land, they  were  yet  very  different  both  in  nature  and  talents. 

Cayley  was  patient  and  equable ;  Sylvester,  fiery  and  passion- 
ate. Cayley  finished  off  a  mathematical  memoir  with  the  same 
care  as  a  legal  instrument;  Sylvester  never  wrote  a  paper  with- 
out foot-notes,  appendices,  supplements;  and  the  alterations 
and  corrections  in  his  proofs  were  such  that  the  printers  found 
their  task  weli-nigh  impossible.  Cayley  was  well-read  in  con- 
temporary mathematics,  and  did  much  useful  work  as  referee 
for  scientific  societies;  Sylvester  read  only  what  had  an  immedi- 
ate bearing  on  his  own  researches,  and  did  little,  if  any,  work 
as  a  referee.  Cayley  was  a  man  of  sound  sense,  and  of  great 
service  in  University  administration;  Sylvester  satisfied  the 
popular  idea  of  a  mathematician  as  one  lost  in  reflection,  and 
high  above  mundane  affairs.  Cayley  was  modest  and  retiring; 
Sylvester,  courageous  and  full  of  his  own  importance.  But 
while  Cayley's  papers,  almost  all,  have  the  stamp  of  pure  logical 
mathematics,  Sylvester's  are  full  of  human  interest.  Cayley 
was  no  orator  and  no  poet;  Sylvester  was  an  orator,  and  if 
not  a  poet,  he  at  least  prided  himself  on  his  poetry.  It  was 
not  long  before  Cayley  was  provided  with  a  chair  at  Cam- 
bridge, where  he  immediately  married,  and  settled  down  to 
work  as  a  mathematician  in  the  midst  of  the  most  favorable 
environment.  Sylvester  was  obliged  to  continue  what  he  called 
"  fighting  the  world  "  alone  and  unmarried. 

There  is  an  ancient  foundation  in  London,  named  after  its 
founder,  Gresham  College.  In  1854  the  lectureship  of  geometry 
fell  vacant  and  Sylvester  applied.  The  trustees  requested  him 
and  I  suppose  also  the  other  candidates,  to  deliver  a  probation- 
ary lecture;  with  the  result  that  he  was  not  appointed.  The 
professorship  of  mathematics  in  the  Royal  Military  Academy 


110  TEN   BRITISH   MATHEMATICIANS 

at  Woolwich  fell  vacant;  Sylvester  was  again  unsuccessful;  but 
the  appointee  died  in  the  course  of  a  year,  and  then  Sylvester 
succeeded  on  a  second  application.  This  was  in  1855,  when  he 
was  41  years  old. 

He  was  a  professor  at  the  Military  Academy  for  fifteen  years; 
and  these  years  constitute  the  period  of  his  greatest  scientific 
activity.  In  addition  to  continuing  his  work  on  the  theory  of 
invariants,  he  was  guided  by  it  to  take  up  one  of  the  most 
difficult  questions  in  the  theory  of  numbers.  Cay  ley  had 
reduced  the  problem  of  the  enumeration  of  invariants  to  that 
of  the  partition  of  numbers;  Sylvester  may  be  said  to  have 
revolutionized  this  part  of  mathematics  by  giving  a  complete 
analytical  solution  of  the  problem,  which  was  in  effect  to 
enumerate  the  solutions  in  positive  integers  of  the  indeterminate 

equation : 

ax+by+cz+  ....  -\-ld  =  m. 

Thereafter  he  attacked  the  similar  problem  connected  with  two 
such  simultaneous  equations  (known  to  Euler  as  the  problem 
of  the  Virgins)  and  was  partially  and  considerably  successful.  In 
June,  1859,  he  delivered  a  series  of  seven  lectures  on  compound 
partition  in  general  at  King's  College,  London.  The  outlines 
of  these  lectures  have  been  published  by  the  Mathematical 
Society  of  London. 

Five  years  later  (1864)  he  contributed  to  the  Royal  Society 
of  London  what  is  considered  his  greatest  mathematical  achieve- 
mant.  Newton,  in  his  lectures  on  algebra,  which  he  called 
"  Universal  Arithmetic  "  gave  a  rule  for  calculating  an  inferior 
limit  to  the  number  of  imaginary  roots  in  an  equation  of  any 
degree,  but  he  did  not  give  any  demonstration  or  indication 
of  the  process  by  which  he  reached  it.  Many  succeeding 
mathematicians  such  as  Euler,  Waring,  Maclaurin,  took  up  the 
problem  'Of  investigating  the  rule,  but  they  were  unable  to  estab- 
lish either  its  truth  or  inadequacy.  Sylvester  in  the  paper 
quoted  established  the  validity  of  the  rule  for  algebraic  equa- 
tions as  far  as  the  fifth  degree  inclusive.  Next  year  in  a  com- 
munication to  the  Mathematical  Society  of  London,  he  fully 
established  and  generalized  the  rule.  "  I  owed  my  success,"  he 


JAMES  JOSEPH   SYLVESTER  111 

said,  ''  chiefly  to  merging  the  theorem  to  be  proved  in  one  of 
greater  scope  and  generality.  In  mathematical  research,  revers- 
ing the  axiom  of  Euclid  and  controverting  the  proposition  of 
Hesiod,  it  is  a  continual  matter  of  experience,  as  I  have  found 
myself  over  and  over  again,  that  the  whole  is  less  than  its  part." 

Two  years  later  he  succeeded  De  Morgan  as  president  of 
the  London  Mathematical  Society.  He  was  the  first  mathe- 
matician to  whom  that  Society  awarded  the  Gold  medal  founded 
in  honor  of  De  Morgan.  In  1869,  when  the  British  Association 
met  in  Exeter,  Prof.  Sylvester  was  president  of  the  section  of 
mathematics  and  physics.  Most  of  the  mathematicians  who 
have  occupied  that  position  have  experienced  difficulty  in  find- 
ing a  subject  which  should  satisfy  the  two  conditions  of  being 
first,  cognate  to  their  branch  of  science;  secondly,  interesting 
to  an  audience  of  general  culture.  Not  so  Sylvester.  He  took 
up  certain  views  of  the  nature  of  mathematical  science  which 
Huxley  the  great  biologist  had  just  published  in  Macmillari's 
Magazine  and  the  Fortnightly  Review.  He  introduced  his  subject 
by  saying  that  he  was  himself  like  a  great  party  leader  and 
orator  in  the  House  of  Lords,  who,  when  requested  to  make  a 
speech  at  some  religious  or  charitable,  at-all-events  non-political 
meeting  declined  the  honor  on  the  ground  that  he  could  not 
speak  unless  he  saw  an  adversary  before  him.  I  shall  now 
quote  from  the  address,  so  that  you  may  hear  Sylvester's  own 
words. 

"  In  obedience,"  he  said,  "  to  a  somewhat  similar  combative 
instinct,  I  set  to  myself  the  task  of  considering  certain  utter- 
ances of  a  most  distinguished  member  of  the  Association,  one 
whom  I  no  less  respect  for  his  honesty  and  public  spirit,  than 
I  admire  for  his  genius  and  eloquence,  but  from  whose  opinions 
on  a  subject  he  has  not  studied  I  feel  constrained  to  differ.  I 
have  no  doubt  that  had  my  distinguished  friend,  the  probable 
president-elect  of  the  next  meeting  of  the  Association,  applied 
his  uncommon  powers  of  reasoning,  induction,  comparison, 
observation  and  invention  to  the  study  of  mathematical  science, 
he  would  have  become  as  great  a  mathematician  as  he  is  now 
a  biologist;  indeed  he  has  given  public  evidence  of  his  ability 


112  TEN   BRITISH  MATHEMATICIANS 

to  grapple  with  the  practical  side  of  certain  mathematical  ques- 
tions; but  he  has  not  made  a  study  of  mathematical  science 
as  such,  and  the  eminence  of  his  position,  and  the  weight  justly 
attaching  to  his  name,  render  it  only  the  more  imperative  that 
any  assertion  proceeding  from  such  a  quarter,  which  may  appear 
to  be  erroneous,  or  so  expressed  as  to  be  conducive  to  error? 
should  not  remain  unchallenged  or  be  passed  over  in  silence. 

"  Huxley  says  '  mathematical  training  is  almost  purely 
deductive.  The  mathematician  starts  with  a  few  simple  prop- 
ositions, the  proof  of  which  is  so  obvious  that  they  are  called 
self-evident,  and  the  rest  of  his  work  consists  of  subtle  deduc- 
tions from  them.  The  teaching  of  languages  at  any  rate  as 
ordinarily  practised,  is  of  the  same  general  nature — authority 
and  tradition  furnish  the  data,  and  the  mental  operations  are 
deductive.'  It  would  seem  from  the  above  somewhat  singularly 
juxtaposed  paragraphs,  that  according  to  Prof.  Huxley,  the 
business  of  the  mathematical  student  is,  from  a  limited  number 
of  propositions  (bottled  up  and  labelled  ready  for  use)  to  deduce 
any  required  result  by  a  process  of  the  same  general  nature 
as  a  student  of  languages  employs  in  declining  and  conjugating 
his  nouns  and  verbs — that  to  make  out  a  mathematical  propo- 
sition and  to  construe  or  parse  a  sentence  are  equivalent  or 
identical  mental  operations.  Such  an  opinion  scarcely  seems 
to  need  serious  refutation.  The  passage  is  taken  from  an  article 
in  Macmillan's  Magazine  for  June  last,  entitled,  '  Scientific 
Education — Notes  of  an  after-dinner  speech';  and  I  cannot 
but  think  would  have  been  couched  in  more  guarded  terms  by 
my  distinguished  friend,  had  his  speech  been  made  before  dinner 
instead  of  after. 

"  The  notion  that  mathematical  truth  rests  on  the  narrow 
basis  of  a  limited  number  of  elementary  propositions  from 
which  all  others  are  to  be  derived  by  a  process  of  logical  inference 
and  verbal  deduction  has  been  stated  still  more  strongly  and 
explicitly  by  the  same  eminent  writer  in  an  article  of  even  date 
with  the  preceeding  in  the  Fortnightly  Review]  where  we  are 
told  that  '  Mathematics  is  that  study  which  knows  nothing  of 
observation,  nothing  of  experiment,  nothing  of  induction,  nothing 


JAMES  JOSEPH  SYLVESTER  113 

of  causation.'  I  think  no  statement  could  have  been  made 
more  opposite  to  the  undoubted  facts  of  the  case,  which  are 
that  mathematical  analysis  is  constantly  invoking  the  aid  of 
new  principles,  new  ideas  and  new  methods  not  capable  of 
being  denned  by  any  form  of  words,  but  springing  direct  from 
the  inherent  powers  and  activity  of  the  human  mind,  and  from 
continually  renewed  introspection  of  that  inner  world  of  thought 
of  which  the  phenomena  are  as  varied  and  require  as  close 
attention  to  discern  as  those  of  the  outer  physical  world;  that 
it  is  unceasingly  calling  forth  the  faculties  of  observation  and 
comparison;  that  one  of  its  principal  weapons  is  induction; 
that  is  has  frequent  recourse  to  experimental  trial  and  veri- 
fication; and  that  it  affords  a  boundless  scope  for  the  exercise 
of  the  highest  efforts  of  imagination  and  invention." 

Huxley  never  replied;  convinced  or  not,  he  had  sufficient 
sagacity  to  see  that  he  had  ventured  far  beyond  his  depth. 
In  the  portion  of  the  address  quoted,  Sylvester  adds  paren- 
thetically a  clause  which  expresses  his  theory  of  mathematical 
knowledge.  He  says  that  the  inner  world  of  thought  in  each 
individual  man  (which  is  the  world  of  observation  to  the  mathe- 
matician) may  be  conceived  to  stand  in  somewhat  the  same 
general  relation  of  correspondence  to  the  outer  physical  world 
as  an  object  to  the  shadow  projected  from  it.  To  him  the 
mental  order  was  more  real  than  the  world  of  sense,  and  the 
foundation  of  mathematical  science  was  ideal,  not  experimental. 

By  this  time  Sylvester  had  received  most  of  the  high  dis- 
tinctions, both  domestic  and  foreign,  which  are  usually  awarded 
to  a  mathematician  of  the  first  rank  in  his  day.  But  a  dis- 
continuity was  at  hand.  The  War  Office  issued  a  regulation 
whereby  officers  of  the  army  were  obliged  to  retire  on  half  pay 
on  reaching  the  age  of  55  years.  Sylvester  was  a  professor  in 
a  Military  College;  in  a  few.  months,  on  his  reaching  the  pre- 
scribed age,  he  was  retired  on  half  pay.  He  felt  that  though 
no  longer  fit  for  the  field  he  was  still  fit  for  the  classroom.  And 
he  felt  keenly  the  diminution  in  his  income.  It  was  about  this 
time  that  he  issued  a  small  volume — the  only  book  he  ever 
published;  not  on  mathematics,  as  you  may  suppose,  but 


114  TEN  BRITISH   MATHEMATICIANS 

entitled  The  Laws  of  Verse.  He  must  have  prided  himself  a 
good  deal  on  this  composition,  for  one  of  his  last  letters  in  Nature 
is  signed  "  J.  J.  Sylvester,  author  of  The  Laws  of  Verse."  He 
made  some  excellent  translations  from  Horace  and  from  German 
poets;  and  like  Sir  W.  R.  Hamilton  he  was  accustomed  to  express 
his  feelings  in  sonnets. 

The  break  in  his  life  appears  to  have  discouraged  Sylvester 
for  the  time  being  from  engaging  in  any  original  research.  But 
after  three  years  a  Russian  mathematician  named  Tschebicheff, 
a  professor  in  the  University  of  Saint  Petersburg,  visiting 
Sylvester  in  London,  drew  his  attention  to  the  discovery  by  a 
Russian  student  named  Lipkin,  of  a  mechanism  for  drawing 
a  perfect  straight  line.  Mr.  Lipkin  received  from  the  Russian 
Government  a  substantial  award.  It  was  found  that  the  same 
discovery  had  been  made  several  years  before  by  M.  Peaucellier, 
an  officer  in  the  French  army,  but  failing  to  be  recognized  at 
its  true  value  had  dropped  into  oblivion.  Sylvester  introduced 
the  subject  into  England  in  the  form  of  an  evening  lecture  before 
the  Royal  Institution,  entitled  "  On  recent  discoveries  in  mechan- 
ical conversion  of  motion."  The  Royal  Institution  of  London 
was  founded  to  promote  scientific  research;  its  professors  have 
been  such  men  as  Davy,  Faraday,  Tyndall,  Dewar.  It  is  not 
a  teaching  institution,  but  it  provides  for  special  courses  of 
lectures  in  the  afternoons  and  for  Friday  evening  lectures  by 
investigators  of  something  new  in  science.  The  evening  lec- 
tures are  attended  by  fashionable  audiences  of  ladies  and  gentle- 
men in  full  dress. 

Euclid  bases  his  Elements  on  two  postulates;  first,  that  a 
straight  line  can  be  drawn,  second,  that  a  circle  can  be  described. 
It  is  sometimes  expressed  in  this  way;  he  postulates  a  ruler  and 
compass.  The  latter  contrivance  is  not  difficult  to  construct, 
because  it  does  not  involve  the  use  of  a  ruler  or  a  compass  in 
its  own  construction.  But  how  is  a  ruler  to  be  made  straight, 
unless  you  already  have  a  ruler  by  which  to  test  it?  The 
problem  is  to  devise  a  mechanism  which  shall  assume  the  second 
postulate  only,  and  be  able  to  satisfy  the  first.  It  is  the  mechan- 
ical problem  of  converting  motion  in  a  circle  into  motion  in 


JAMES  JOSEPH  SYLVESTER  115 

a  straight  line,  without  the  use  of  any  guide.  James  Watt, 
the  inventor  of  the  steam-engine,  tackled  the  problem  with  all 
his  might,  but  gave  it  up  as  impossible.  However,  he  succeeded 
in  finding  a  contrivance  which  solves  the  problem  very  approxi- 
mately. Watt's  parallelogram,  employed  in  nearly  every  beam- 
engine,  consists  of  three  links;  of  which  AC  and  BD  are 
equal,  and  have  fixed  pivots  at  A  and  B  respectively.  The 
link  CD  is  of  such  a  length  that  AC  and  BD  are  parallel 
when  horizontal.  The  tracing 

AO "c 

point  is  attached  to  the  middle 
point  of  CD.    When  C  and  D 


move  round  their  pivots,  the 

tracing  point  describes  a  straight  line  very  approximately,  so 
long  as  the  arc  of  displacement  is  small.  The  complete  figure 
which  would  be  described  is  the  figure  of  8,  and  the  part  utilized 
is  near  the  point  of  contrary  flexure. 

A  linkage  giving  a  closer  approximation  to  a  straight  line 
was  also  invented  by  the  Russian  mathematician  before  men- 
tioned— Tschebicheff;  it  likewise  made  use  of  three  links.  But 
the  linkage  invented  by  Peaucellier  and  later  by  Lipkin  had 
seven  pieces.  The  arms  AB  and  AC  are  of  equal  length,  and 

have  a  fixed  pivot  at  A.  The  links 
DB,  BE,  EC,  CD  are  of  equal  length. 
EF  is  an  arm  connecting  E  with  the 
fixed  pivot  F  and  is  equal  in  length  to 
the  distance  between  A  and  F.  It  is 
readily  shown  by  geometry  that,  as 
the  point  E  describes  a  circle  around 
the  center  F,  the  point  D  describes 
an  exact  straight  line  perpendicular  to  the  line  joining  it  and  F. 
The  exhibition  of  this  contrivance  at  work  was  the  climax  of 
Sylvester's  lecture. 

In  Sylvester's  audience  were  two  mathematicians,  Hart  and 
Kempe,  who  took  up  the  subject  for  further  investigation.  Hart 
perceived  that  the  contrivances  of  Watt  and  of  Tschebicheff 
consisted  of  three  links,  whereas  Peaucellier's  consisted  of  seven. 
Accordingly  he  searched  for  a  contrivance  of  five  links  which 


116  TEN  BRITISH  MATHEMATICIANS 

would  enable  a  tracing  point  to  describe  a  perfect  straight  line; 
and  he  succeeded  in  inventing  it.  Kempe  was  a  London  barrister 
whose  specialty  was  ecclesiastical  law.  He  and  Sylvester  worked 
up  the  theory  of  linkages  together,  and  discovered  among  other 
things  the  skew  pantograph.  Kempe  became  so  imbued  with 
linkage  that  he  contributed  to  the  Royal  Society  of  London 
a  paper  on  the  "  Theory  of  Mathematical  Form,"  in  which  he 
explains  all  reasoning  by  means  of  linkages. 

About  this  time  (1877)  the  Johns  Hopkins  University  was 
organized  at  Baltimore,  and  Sylvester,  at  the  age  of  63,  was 
appointed  the  first  professor  of  mathematics.  Of  his  work 
there  as  a  teacher,  one  of  his  pupils,  Dr.  Fabian  Franklin,  thus 
spoke  in  an  address  delivered  at  a  memorial  meeting  in  that 
University:  "  The  one  thing  which  constantly  marked  Syl- 
vester's lectures  was  enthusiastic  love  of  the  thing  he  was  doing. 
He  had  in  the  fullest  possible  degree,  to  use  the  French  phrase, 
the  defect  of  this  quality;  for  as  he  almost  always  spoke  with 
enthusiastic  ardor,  so  it  was  almost  never  possible  for  him  to 
speak  on  matters  incapable  of  evoking  this  ardor.  In  other 
words,  the  substance  of  his  lectures  had  to  consist  largely  of  his 
own  work,  and,  as  a  rule,  of  work  hot  from  the  forge.  The 
consequence  was  that  a  continuous  and  systematic  presentation 
of  any  extensive  body  of  doctrine  already  completed  was  not  to 
be  expected  from  him.  Any  unsolved  difficulty,  any  suggested 
extension,  such  as  would  have  been  passed  by  with  a  mention 
by  other  lecturers,  became  inevitably  with  him  the  occasion  of 
a  digression  which  was  sure  to  consume  many  weeks,  if  indeed 
it  did  not  take  him  away  from  the  original  object  permanently. 
Nearly  all  of  the  important  memoirs  which  he  published,  while 
in  Baltimore,  arose  in  this  way.  We  who  attended  his  lectures 
may  be  said  to  have  seen  these  memoirs  in  the  making.  He 
would  give  us  on  the  Friday  the  outcome  of  his  grapplings  with 
the  enemy  since  the  Tuesday  lecture.  Rarely  can  it  have 
fallen  to  the  lot  of  any  class  to  follow  so  completely  the  workings 
of  the  mind  of  the  master.  Not  only  were  all  thus  privileged  to 
see  '  the  very  pulse  of  the  machine,'  to  learn  the  spring  and 
motive  of  the  successive  steps  that  led  to  his  results,  but  we 


JAMES  JOSEPH  SYLVESTER  117 

were  set  aglow  by  the  delight  and  admiration  which,  with  perfect 
naivete  and  with  that  luxuriance  of  language  peculiar  to  him, 
Sylvester  lavished  upon  these  results.  That  in  this  enthusiastic 
admiration  he  sometimes  lacked  the  sense  of  proportion  cannot 
be  denied.  A  result  announced  at  one  lecture  and  hailed  with 
loud  acclaim  as  a  marvel  of  beauty  was  by  no  means  sure  of 
not  being  found  before  the  next  lecture  to  have  been  erroneous; 
but  the  Esther  that  supplanted  this  Vashti  was  quite  certain 
to  be  found  still  more  supremely  beautiful.  The  fundamental 
thing,  however,  was  not  this  occasional  extravagance,  but  the 
deep  and  abiding  feeling  for  truth  and  beauty  which  underlay 
it.  No  young  man  of  generous  mind  could  stand  before  that 
superb  grey  head  and  hear  those  expositions  of  high  and  dear- 
bought  truths,  testifying  to  a  passionate  devotion  undimmed 
by  years  or  by  arduous  labors,  without  carrying  away  that 
which  ever  after  must  give  to  the  pursuit  of  truth  a  new  and 
deeper  significance  in  his  mind." 

One  of  Sylvester's  principal  achievements  at  Baltimore  was 
the  founding  of  the  American  Journal  of  Mathematics,  which, 
at  his  suggestion,  took  the  quarto  form.  He  aimed  at  estab- 
lishing a  mathematical  journal  in  the  English  language,  which 
should  equal  Liouville's  Journal  in  France,  or  Crelle's  Journal 
in  Germany.  Probably  his  best  contribution  to  the  American 
Journal  consisted  in  his  "  Lectures  on  Universal  Algebra  "; 
which,  however,  were  left  unfinished,  like  a  great  many  other 
projects  of  his. 

Sylvester  had  that  quality  of  absent-mindedness  which  is 
popularly  supposed  to  be,  if  not  the  essence,  at  least  an  invariable 
accompaniment,  of  a  distinguished  mathematician.  Many 
stories  are  related  on  this  point,  which,  if  not  all  true,  are  at 
least  characteristic.  Dr.  Franklin  describes  an  instance  which 
actually  happened  in  Baltimore.  To  illustrate  a  theory  of 
versification  contained  in  his  book  The  Laws  of  Verse,  Sylvester 
prepared  a  poem  of  400  lines,  ah1  rhyming  with  the  name  Rosa- 
lind or  Rosalind;  and  it  was  announced  that  the  professor  would 
read  the  poem  on  a  specified  evening  at  a  specified  hour  at  the 
Peabody  Institute.  At  the  time  appointed  there  was  a  large 


118  TEN   BRITISH   MATHEMATICIANS 

turn-out  of  ladies  and  gentlemen.  Prof.  Sylvester,  as  usual, 
had  a  number  of  footnotes  appended  to  his  production;  and  he 
announced  that  in  order  to  save  interruption  in  reading  the 
poem  itself,  he  would  first  read  the  footnotes.  The  reading  of 
the  footnotes  suggested  various  digressions  to  his  imagination; 
an  hour  had  passed,  still  no  poem;  an  hour  and  a  half  passed 
and  the  striking  of  the  clock  or  the  unrest  of  his  audience 
reminded  him  of  the  promised  poem.  He  was  astonished  to 
find  how  time  had  passed,  excused  all  who  had  engagements, 
and  proceeded  to  read  the  Rosalind  poem. 

In  the  summer  of  1881  I  visited  London  to  see  the  Electrical 
Exhibition  in  the  Crystal  Palace — one  of  the  earliest  exhibitions 
devoted  to  electricity  exclusively.  I  had  made  some  investi- 
gations on  the  electric  discharge,  using  a  Holtz  machine  where 
De  LaRue  used  a  large  battery  of  cells.  Mr.  De  LaRue  was 
Secretary  of  the  Royal  Institution;  he  gave  me  a  ticket  to  a 
Friday  evening  discourse  to  be  delivered  by  Mr.  Spottiswoode, 
then  president  of  the  Royal  Society,  on  the  phenomena  of  the 
intensive  discharge  of  electricity  through  gases;  also  an  invi- 
tation to  a  dinner  at  his  own  house  to  be  given  prior  to  the 
lecture.  Mr.  Spottiswoode,  the  lecturer  for  the  evening,  was 
there;  also  Prof.  Sylvester.  He  was  a  man  rather  under  the 
average  height,  with  long  gray  beard  and  a  profusion  of  gray 
locks  round  his  head  surmounted  by  a  great  dome  of  forehead. 
He  struck  me  as  having  the  appearance  of  an  artist  or  a  poet 
rather  than  of  an  exact  scientist.  After  dinner  he  conversed 
very  eloquently  with  an  elderly  lady  of  title,  while  I  conversed 
with  her  daughter.  Then  cabs  were  announced  to  take  us  to 
the  Institution.  Prof.  Sylvester  and  I,  being  both  bachelors, 
were  put  in  a  cab  together.  The  professor,  who  had  been  so 
eloquent  with  the  lady,  said  nothing;  so  I  asked  him  how  he 
liked  his  work  at  the  Johns  Hopkins  University.  "  It  is  very 
pleasant  work  indeed,"  said  he,  "  and  the  young  men  who 
study  there  are  all  so  enthusiastic."  We  had  not  exhausted 
that  subject  before  we  reached  our  destination.  We  went  up 
the  stairway  together,  then  Sylvester  dived  into  the  library  to 
see  the  last  number  of  Comptes  Rendus  (in  which  he  published 


JAMES   JOSEPH   SYLVESTER 


119 


many  of  his  results  at  that  time)  and  I  saw  him  no  more.  I  have 
always  thought  it  very  doubtful  whether  he  came  out  to  hear 
Spottiswoode's  lecture. 

We  have  seen  that  H.  J.  S.  Smith,  the  Savilian  professor  of 
Geometry  at  Oxford,  died  in  1883.  Sylvester's  friends  urged 
his  appointment,  with  the  result  that  he  was  elected.  After  two 
years  he  delivered  his  inaugural  lecture;  of  which  the  subject 
was  differential  invariants,  termed  by  him  reciprocants.  An 

d  v  d  v 

elementary  reciprocant  is  -~,  for  if  -7^  = 


then  —  =  o. 


He 


looked  upon  this  as  the 
the  "  chrysalis  " 


grub  "  form,  and  developed  from  it 


dx2 
d2<j> 


d2$ 
dxdy 


dxdy         dy2 


dx 


and  the  "  imago 


dy 


d2<&          d2<f> 

dx2 
d2® 

dxdy 
d2* 

dxdy 

d2* 
dxdr 

dy2 

dydr 

dx' 


dy' 


d2® 

dxdr' 


dydr' 

d2* 
dr2' 

You  will  observe  that  the  chrysalis  expression  is  unsymmetrical; 
the  place  of  a  ninth  term  is  vacant.  It  moved  Sylvester's  poetic 
imagination,  and  into  his  inaugural  lecture  he  interjected  the 
following  sonnet: 

To  A  MISSING  MEMBER  OF  A  FAMILY  GROUP  OF  TERMS  IN 
AN  ALGEBRAICAL  FORMULA: 

Lone  and  discarded  one !  divorced  by  fate, 
Far  from  thy  wished-for  fellows — whither  art  flown? 
Where  lingerest  thou  in  thy  bereaved  estate, 
Like  some  lost  star,  or  buried  meteor  stone? 


120  TEN  BRITISH  MATHEMATICIANS 

Thou  minds't  me  much  of  that  presumptuous  one, 

Who  loth,  aught  less  than  greatest,  to  be  great; 

From  H.aven's  immensity  fell  headlong  down 

To  live  forlorn,  self-centred,  desolate: 

Or  who,  new  Heraklid,  hard  exile  bore, 

Now  buoyed  by  hope,  now  stretched  on  rack  of  fear, 

Till  throned  Astraea,  wafting  to  his  ear 

Words  of  dim  portent  through  the  Atlantic  roar, 

Bade  him  "  the  sanctuary  of  the  Muse  revere 

And  strew  with  flame  the  dust  of  Isis'  shore." 

This  inaugural  lecture  was  the  beginning  of  his  last  great 
contribution  to  mathematics,  and  the  subsequent  lectures  of 
that  year  were  devoted  to  his  researches  in  that  line.  Smith 
and  Sylvester  were  akin  in  devoting  attention  to  the  theory  of 
numbers,  and  also  in  being  eloquent  speakers.  But  in  other 
respects  the  Oxonians  found  a  great  difference.  Smith  had 
been  a  painstaking  tutor;  Sylvester  could  lecture  only  on  his 
own  researches,  which  were  not  popular  in  a  place  so  wholly 
given  over  to  examinations.  Smith  was  an  incessantly  active 
man  of  affairs;  Sylvester  became  the  subject  of  melancholy 
and  complained  that  he  had  no  friends. 

In  1872  a  deputy  professor  was  appointed.  Sylvester  re- 
moved to  London,  and  lived  mostly  at  the  Athenaeum  Club. 
He  was  now  78  years  of  age,  and  suffered  from  partial  loss  of 
sight  and  memory.  He  was  subject  to  melancholy,  and  his 
condition  was  indeed  "  forlorn  and  desolate."  His  nearest  rela- 
tives were  nieces,  but  he  did  not  wish  to  ask  their  assistance. 
One  day,  meeting  a  mathematical  friend  who  had  a  home  in  Lon- 
don, he  complained  of  the  fare  at  the  Club,  and  asked  his  friend 
to  help  him  find  suitable  private  apartments  where  he  could  have 
better  cooking.  They  drove  about  from  place  to  place  for  a 
whole  afternoon,  but  none  suited  Sylvester.  It  grew  late:  Syl- 
vester said,  "  You  have  a  pleasant  home:  take  me  there,"  and 
this  was  done.  Arrived,  he  appointed  one  daughter  his  reader 
and  another  daughter  his  amanuensis.  "  Now,"  said  he,  "I  feel 
comfortably  installed;  don't  let  my  relatives  know  where  I 
am."  The  fire  of  his  temper  had  not  dimmed  with  age,  and  it 
required  all  the  Christian  fortitude  of  the  ladies  to  stand  his 


JAMES  JOSEPH   SYLVESTER  121 

exactions.  Eventually,  notice  had  to  be  sent  to  his  nieces  to 
come  and  take  charge  of  him.  He  died  on  the  i5th  of  March, 
1897,  in  the  83d  year  of  his  age,  and  was  buried  in  the  Jewish 
cemetery  at  Dalston. 

As  a  theist,  Sylvester  did  not  approve  of  the  destructive 
attitude  of  such  men  as  Clifford,  in  matters  of  religion.  In  the 
early  days  of  his  career  he  suffered  much  from  the  disabilities 
attached  to  his  faith,  and  they  were  the  prime  cause  of  so  much 
"  fighting  the  world."  He  was,  in  all  probability,  a  greater 
mathematical  genius  than  Cayley;  but  the  environment  in 
which  he  lived  for  some  years  was  so  much  less  favorable  that 
he  was  not  able  to  accomplish  an  equal  amount  of  solid  work. 
Sylvester's  portrait  adorns  St.  John's  College,  Cambridge.  A 
memorial  fund  of  £1500  has  been  placed 'in  the  charge  of  the 
Royal  Society  of  London,  from  the  proceeds  of  which  a  medal 
and  about  £100  in  money  is  awarded  triennially  for  work 
done  in  pure  mathematics.  The  first  award  has  been  made  to 
M.  Henri  Poincare  of  Paris,  a  mathematician  for  whom  Sylvester 
had  a  high  professional  and  personal  regard. 


THOMAS  PENYNGTON   KIRKMAN* 

(1806-1895) 

THOMAS  PENYNGTON  KIRKMAN  was  born  on  March  31,  1806, 
at  Bolton  in  Lancashire.  He  was  the  son  of  John  Kirkman, 
a  dealer  in  cotton  and  cotton  waste;  he  had  several  sisters  but 
no  brother.  He  was  educated  at  the  Grammar  School  of  Bolton, 
where  the  tuition  was  free.  There  he  received  good  instruction 
in  Latin  and  Greek,  but  no  instruction  in  geometry  or  algebra; 
even  Arithmetic  was  not  then  taught  in  the  headmaster's  upper 
room.  He  showed  a  decided  taste  for  study  and  was  by  far 
the  best  scholar  in  the  school.  His  father,  who  had  no  taste 
for  learning  and  was  succeeding  in  trade,  was  determined  that 
his  only  son  should  follow  his  own  business,  and  that  without 
any  loss  of  time.  The  schoolmaster  tried  to  persuade  the  father 
to  let  his  son  remain  at  school;  and  the  vicar  also  urged  the 
father,  saying  that  if  he  would  send  his  son  to  Cambridge  Uni- 
versity, he  would  guarantee  for  sixpence  that  the  boy  would 
win  a  fellowship.  But  the  father  was  obdurate;  young  Kirk- 
man was  removed  from  school,  when  he  was  fourteen  years  of 
age,  and  placed  at  a  desk  in  his  father's  office.  While  so  engaged, 
he  continued  of  his  own  accord  his  study  of  Latin  and  Greek, 
and  added  French  and  German. 

After  ten  years  spent  in  the  counting  room,  he  tore  away 
from  his  father,  secured  the  tuition  of  a  young  Irish  baronet, 
Sir  John  Blunden,  and  entered  the  University  of  Dublin  with 
the  view  of  passing  the  examinations  for  the  degree  of  B.A. 
There  he  never  had  instruction  from  any  tutor.  It  was  not 
until  he  entered  Trinity  College,  Dublin,  that  he  opened  any 
mathematical  book.  He  was  not  of  course  abreast  with  men 
who  had  good  preparation.  What  he  knew  of  mathematics, 

*  This  Lecture  was  delivered  April  20,  1903. — EDITORS. 
122 


THOMAS  PENYNGTON   KIRKMAN  123 

he  owed  to  his  own  study,  having  never  had  a  single  hour's 
instruction  from  any  person.  To  this  self-education  is  due,  it 
appears  to  me,  both  the  strength  and  the  weakness  to  be  found 
in  his  career  as  a  scientist.  However,  in  his  college  course  he 
obtained  honors,  or  premiums  as  they  are  called,  and  graduated 
as  a  moderator,  something  like  a  wrangler. 

Returning  to  England  in  1835,  when  he  was  29  years  old, 
he  was  ordained  as  a  minister  in  the  Church  of  England.  He 
was  a  curate  for  five  years,  first  at  Bury,  afterwards  at  Lymm; 
then  he  became  the  vicar  of  a  newly-formed  parish — Croft  with 
Southworth  in  Lancashire.  This  parish  was  the  scene  of  his 
life's  labors.  The  income  of  the  benefice  was  not  large,  about 
£200  per  annum;  for  several  years  he  supplemented  this  by 
taking  pupils.  He  married,  and  property  which  came  to  his 
wife  enabled  them  to  dispense  with  the  taking  of  pupils.  His 
father  became  poorer,  but  was  able  to  leave  some  property  to 
his  son  and  daughters.  His  parochial  work,  though  small,  was 
discharged  with  enthusiasm;  out  of  the  roughest  material  he 
formed  a  parish  choir  of  boys  and  girls  who  could  sing  at  sight 
any  four-part  song  put  before  them.  After  the  private  teach- 
ing was  over  he  had  the  leisure  requisite  for  the  great  mathe- 
matical researches  in  which  he  now  engaged. 

Soon  after  Kirkman  was  settled  at  Croft,  Sir  William  Rowan 
Hamilton  began  to  publish  his  quaternion  papers  and,  being  a 
graduate  of  Dublin  University,  Kirkman  was  naturally  one  of 
the  first  to  study  the  new  analysis.  As  the  fruit  of  his  medi- 
tations he  contributed  a  paper,  to  the  Philosophical  Magazine 
"  On  pluquaternions  and  homoid  products  of  sums  of  n  squares." 
He  proposed  the  appellation  "  pluquaternions  "  for  a  linear 
expression  involving  more  than  three  imaginaries  (the  i,  j,  k  of 
Hamilton),  "  not  dreading  "  he  says,  "  the  pluperfect  criticism 
of  grammarians,  since  the  convenient  barbarism  is  their  own." 
Hamilton,  writing  to  De  •  Morgan,  remarked  "  Kirkman  is  a 
very  clever  fellow,"  where  the  adjective  has  not  the  American 
colloquial  meaning  but  the  English  meaning. 

For  his  own  education  and  that  of  his  pupils  he  devoted 
much  attention  to  mathematical  mnemonics,  studying  the 


124  TEN   BRITISH   MATHEMATICIANS 

Memoria  Technica  of  Grey.  In  1851  he  contributed  a  paper  on 
the  subject  to  the  Literary  and  Philosophical  Society  of  Man- 
chester, and  in  1852  he  published  a  book,  First  Mnemonical  Les- 
sons in  Geometry,  Algebra,  and  Trigonometry,  which  is  dedicated 
to  his  former  pupil,  Sir  John  Blunden.  De  Morgan  pronounced 
it  "  the  most  curious  crochet  I  ever  saw,"  which  was  saying  a 
great  deal,  for  De  Morgan  was  familiar  with  many  quaint  books 
in  mathematics.  In  the  preface  he  says  that  much  of  the  dis- 
taste for  mathematical  study  springs  largely  from  the  difficulty 
of  retaining  in  the  memory  the  previous  results  and  reasoning. 
"  This  difficulty  is  closely  connected  with  the  unpronounceable- 
ness  of  the  formulae;  the  memory  of  the  tongue  and  the  ear  arc. 
not  easily  turned  to  account;  nearly  everything  depends  on  the 
thinking  faculty  or  on  the  practice  of  the  eye  alone.  Hence 
many,  who  see  hardly  anything  formidable  in  the  study  of  a 
language,  look  upon  mathematical  acquirements  as  beyond 
their  power,  when  in  truth  they  are  very  far  from  being  so. 
My  object  is  to  enable  the  learner  to  '  talk  to  himself,'  in  rapid, 
vigorous  and  suggestive  syllables,  about  the  matters  which  he 
must  digest  and  remember.  I  have  sought  to  bring  the  memory 
of  the  vocal  organs  and  the  ear  to  the  assistance  of  the  reasoning 
faculty  and  have  never  scrupled  to  sacrifice  either  good  grammar 
or  good  English  in  order  to  secure  the  requisites  for  a  useful 
mnemonic,  which  are  smoothness,  condensation,  and  jingle. 

As  a  specimen  of  his  mnemonics  we  may  take  the  cotangent 
formula  in  spherical  trigonometry: 

cot  A  sin  C+cos  b  cos  C  =  cot  a  sin  b 

To  remember  this  formula  most  masters  then  required  some  aid 
to  the  memory ;  for  instance  the  following :  If  in  any  spherical 
triangle  four  parts  be  taken  in  succession,  such  as  AbCa,  consist- 
ing of  two  means  bC  and  two  extrerpes  Aa,  then  the  product  of 
the  cosines  of  the  two  means  is  equal  to  the  sine  of  the  mean 
side  X  cotangent  of  the  extreme  side  minus  sine  of  the  mean 
angle  X  cotangent  of  the  extreme  angle,  that  is 

cos  b  cos  C  =  sin  b  cot  a  —  sin  C  cot  A . 


THOMAS  PENYNGTON   KIRKMAN  125 

This  is  an  appeal  to  the  reason.  Kirkman,  however,  proceeds 
on  the  principle  of  appealing  to  the  memory  of  the  ear,  of  the 
tongue,  and  of  the  lips  altogether;  a  true  memoria  technica. 
He  distinguishes  the  large  letter  from  the  small  by  calling  them 
Ang,  Bang,  Cang  (ang  from  angle  in  contrast  to  side).  To 
make  the  formula  more  euphoneous  he  drops  the  s  from  cos 
and  the  n  from  sin.  Hence  the  formula  is 

cot  Ang  si  Cang  and  co  b  co  Cang  are  cot  a  si  b 

which  is  to  be  chanted  till  it  becomes  perfectly  familiar  to  the 
ear  and  the  lips.  The  former  rule  is  a  hint  offered  to  the  judg- 
ment; Kirkman's  method  is  something  to  be  taught  by  rote. 
In  his  book  Kirkman  makes  much  use  of  verse,  in  the  turning 
of  which  he  was  very  skillful. 

In  the  early  part  of  the  nineteenth  century  a  publication 
named  the  Lady's  and  Gentlemen's  Diary  devoted  several  columns 
to  mathematical  problems.  In  1844  the  editor  offered  a  prize 
for  the  solution  of  the  following  question:  "  Determine  the  num- 
ber of  combinations  that  can  be  made  out  of  n  symbols,  each 
combination  having  p  symbols,  with  this  limitation,  that  no 
combination  of  q  symbols  which  may  appear  in  any  one  of 
them,  may  be  repeated  in  any  other."  This  is  a  problem  of 
great  difficulty;  Kirkman  solved  it  completely  for  the  special 
case  of  ^  =  3  and  q=2  and  printed  his  results  in  the  second 
volume  of  the  Cambridge  and  Dublin  Mathematical  Journal. 
As  a  chip  off  this  work  he  published  in  the  Diary  for  1850  the 
famous  problem  of  the  fifteen  schoolgirls  as  follows:  "Fifteen 
young  ladies  of  a  school  walk  out  three  abreast  for  seven  days 
in  succession;  it  is  required  to  arrange  them  daily  so  that  no  two 
shall  walk  abreast  more  than  once."  To  form  the  schedules 
for  seven  days  is  not  difficult;  but  to  find  all  the  possible 
schedules  is  a  different  matter.  Kirkman  found  all  the  possible 
combinations  of  the  fifteen  young  ladies  in  groups  of  three  to 
be  35,  and  the  problem  was  also  considered  and  solved  by  Cayley, 
and  has  been  discussed  by  many  later  writers;  Sylvester  gave 
91  as  the  greatest  number  of  days;  and  he  also  intimated  that 
the  principle  of  the  puzzle  was  known  to  him  when  an  under- 


126  TEN  BRITISH  MATHEMATICIANS 

graduate  at  Cambridge,  and  that  he  had  given  it  to  fellow 
undergraduates.  Kirkman  replied  that  up  to  the  time  he  pro- 
posed the  problem  he  had  neither  seen  Cambridge  nor  met 
Sylvester,  and  narrated  how  he  had  hit  on  the  question. 

The  Institute  of  France  offered  several  times  in  succession 
a  prize  for  a  memoir  on  the  theory  of  the  polyedra;  this  fact 
together  with  his  work  in  combinations  led  Kirkman  to  take 
up  the  subject.  He  always  writes  polyedron  not  polyhedron; 
for  he  says  we  write  periodic  not  perihodic.  When  Kirkman 
began  work  nothing  had  been  done  beyond  the  very  ancient 
enumeration  of  the  five  regular  solids  and  the  simple  combi- 
nations of  crystallography.  His  first  paper,  "  On  the  represen- 
tation and  enumeration  of  the  polyedra,"  was  communicated 
in  1850  to  the  Literary  and  Philosophical  Society  of  Manchester. 
He  starts  with  the  well-known  theorem  P-\-S  =  L-\-2,  where  P 
is  the  number  of  points  or  summits,  S  the  number  of  plane 
bounding  surfaces  and  L  the  number  of  linear  edges  in  a  geo- 
metrical solid.  "  The  question — how  many  w-edrons  are 
there? — has  been  asked,  but  it  is  not  likely  soon  to  receive  a 
definite  answer.  It  is  far  from  being  a  simple  question,  even 
when  reduced  to  the  narrower  compass — how  many  w-edrons 
are  there  whose  summits  are  all  trihedral  "?  He  enumerated 
and  constructed  the  fourteen  8-edra  whose  faces  are  all  tri- 
angles. 

In  1858  the  French  Institute  modified  its  prize  question. 
As  the  subject  for  the  concours  of  1861  was  announced:  "Per- 
fectionner  en  quelque  point  important  la  theorie  g6ometrique 
des  polyedres,"  where  the  indefiniteness  of  the  question  indi- 
cates the  very  imperfect  state  of  knowledge  on  the  subject. 
The  prize  offered  was  3000  francs.  Kirkman  appears  to  have 
worked  at  it  with  a  view  of  competing,  but  he  did  not  send  in 
his  memoir.  Cayley  appears  to  have  intended  to  compete. 
The  time  was  prolonged  for  a  year,  but  there  was  no  award  and 
the  prize  was  taken  down.  Kirkman  communicated  his  results 
to  the  Royal  Society  through  his  friend  Cayley,  and  was  soon 
elected  a  Fellow.  Then  he  contributed  directly  an  elaborate 
paper  entitled  "  Complete  theory  of  the  Polyedra."  In  the 


THOMAS  PENYNGTON   KIRKMAN  127 

preface  he  says,  "  The  following  memoir  contains  a  complete 
solution  of  the  classification  and  enumeration  of  the  P-edra 
Q-acra.  The  actual  construction  of  the  solids  is  a  task  imprac- 
ticable from  its  magnitude,  but  it  is  here  shown  that  we  can 
enumerate  them  with  an  accurate  account  of  their  symmetry 
to  any  values  of  P  and  Q."  The  memoir  consisted  of  21  sec- 
tions; only  the  two  introductory  sections,  occupying  45  quarto 
pages,  were  printed  by  the  Society,  while  the  others  still  remain 
in  manuscript.  During  following  years  he  added  many  con- 
tributions to  this  subject. 

In  1858  the  French  Academy  also  proposed  a  problem  in  the 
Theory  of  Groups  as  the  subject  for  competition  for  the  grand 
mathematical  prize  in  1860:  "  Quels  peuvent  etre  les  nombres 
de  valeurs  des  f  onctions  bien  definies  qui  contiennent  un  nombre 
donne  de  lettres,  et  comment  peut  on  former  les  fonctions  pour 
lesquelles  il  existe  un  nombre  donne  de  valeurs? "  Three 
memoirs  were  presented,  of  which  Kirkman's  was  one,  but  no 
prize  was  awarded.  Not  the  slightest  summary  was  vouch- 
safed of  what  the  competitors  had  added  to  science,  although 
it  was  confessed  that  all  had  contributed  results  both  new  and 
important;  and  the  question,  though  proposed  for  the  first 
time  for  the  year  1860,  was  withdrawn  from  competition  con- 
trary to  the  usual  custom  of  the  Academy.  Kirkman  con- 
tributed the  results  of  his  investigation  to  the  Manchester 
Society  under  the  title  "  The  complete  theory  of  groups,  being 
the  solution  of  the  mathematical  prize  question  of  the  French 
Academy  for  1860."  In  more  recent  years  the  theory  of  groups 
has  engaged  the  attention  of  many  mathematicians  in  Germany 
and  America;  so  far  as  British  contributors  are  concerned 
Kirkman  was  the  first  and  still  remains  the  greatest. 

In  1 86 1  the  British  Association  met  at  Manchester;  it  was 
the  last  of  its  meetings  which  Sir  William  Rowan  Hamilton 
attended.  After  the  meeting  Hamilton  visited  Kirkman  at  his 
home  in  the  Croft  rectory,  and  that  meeting  was  no  doubt  a 
stimulus  to  both.  As  regards  pure  mathematics  they  were 
probably  the  two  greatest  in  Britain;  both  felt  the  loneliness  of 
scientific  work,  both  were  metaphysicians  of  penetrating  power, 


128  TEN  BRITISH  MATHEMATICIANS 

both  were  good  versifiers  if  not  great  poets.  Of  nearly  the  same 
age,  they  were  both  endowed  with  splendid  physique;  but  the 
care  which  was  taken  of  their  health  was  very  different;  in 
four  years  Hamilton  died  but  Kirkman  lived  more  than  30  years 
longer. 

About  1862  the  Educational  Times,  a  monthly  periodical 
published  in  London,  began  to  devote  several  columns  to  the 
proposing  and  solving  of  mathematical  problems,  taking  up 
the  work  after  the  demise  of  the  Diary.  This  matter  was  after- 
wards reprinted  in  separate  volumes,  two  for  each  year.  In 
these  reprints  are  to  be  found  many  questions  proposed  by 
Kirkman;  they  are  generally  propounded  in  quaint  verse,  and 
many  of  them  were  suggested  by  his  study  of  combinations.  A 
good  specimen  is  "  The  Revenge  of  Old  King  Cole  " 

"  Full  oft  ye  have  had  your  fiddler's  fling, 

For  your  own  fun  over  the  wine; 

And  now  "  quoth  Cole,  the  merry  old  king, 

"  Ye  shall  have  it  again  for  mine. 

My  realm  prepares  for  a  week  of  joy 

At  the  coming  of  age  of  a  princely  boy — 

Of  the  grand  six  days  procession  in  square, 

In  all  your  splendour  dressed, 

Filling  the  city  with  music  rare 

From  fiddlers  five  abreast,"  etc. 

The  problem  set  forth  by  this  and  other  verses  is  that  of 
25  men  arranged  in  five  rows  on  Monday.  Shifting  the  second 
column  one  step  upward,  the  third  two  steps,  the  fourth  three 

Monday  Tuesday  Wednesday  Thursday 


A   B    C    D    E 

A    G   M    S   Y 

A   L    W   I    T 

A   Q    H   X   O 

F    G    H    I     J 

F    L    R    X  E 

F    Q    C    N  Y 

F    V   M   D    T 

K  L   M   N    O 

K   Q   W  D    J 

K   V    H    S  E 

K  B    R    I    Y 

P    Q    R    S    T 

P    V    C    I    0 

P    B   M   X    J 

P   G  W   N   E 

U  V  W  X  Y 

U   B    H   N   T 

N  G    R    D    O 

U  L    C     S    J 

steps,  and  the  fifth  four  steps  gives  the  arrangement  for  Tuesday. 
Applying  the  same  rule  to  Tuesday  gives  Wednesday's  array, 
and  similarly  are  found  those  for  Thursday  and  Friday.  In 


THOMAS  PENYNGTON  KIRKMAN  129 

none  of  these  can  the  same  two  men  be  found  in  one  row.  But 
the  rule  fails  to  work  for  Saturday,  so  that  a  special  arrangement 
must  be  brought  in  which  I  leave  to  my  hearers  to  work  out. 
This  problem  resembles  that  of  the  fifteen  schoolgirls. 

The  Rev.  Kirkman  became  at  an  early  period  of  his  life  a 
broad  churchman.  About  1863  he  came  forward  in  defense  of 
the  Bishop  of  Colenso,  a  mathematician,  and  later  he  contributed 
to  a  series  of  pamphlets  published  in  aid  of  the  cause  of  "Free 
Enquiry  and  Free  Expression."  In  one  of  his  letters  to  me 
Kirkman  writes  as  follows:  "  The  Life  of  Colenso  by  my  friend 
Rev.  Sir  George  Cox,  Bart.,  is  a  most  charming  book;  and 
the  battle  of  the  Bishops  against  the  lawyers  in  the  matter  of 
the  vacant  see  of  Natal,  to  which  Cox  is  the  bishop-elect,  is 
exciting.  Canterbury  refuses  to  ask,  as  required,  the  Queen's 
mandate  to  consecrate  him.  The  Natal  churchmen  have  just 
petitioned  the  Queen  to  make  the  Primate  do  his  duty  accord- 
ing to  law.  Natal  was  made  a  See  with  perpetual  succession, 
and  is  endowed.  The  endowment  has  been  lying  idle  since 
Colenso 's  death  in  1883;  and  the  bishops  who  have  the  law 
courts  dead  against  them  here  are  determined  that  no  successor 
to  Colenso  shall  be  consecrated.  There  is  a  Bishop  of  South 
African  Church  there,  whom  they  thrust  in  while  Colenso  lived, 
on  pretense  that  Colenso  was  excommunicate.  We  shall  soon 
see  whether  the  lawyers  or  the  bishops  are  to  win."  It  was 
Kirkman's  own  belief  that  his  course  in  this  matter  injured 
his  chance  of  preferment  in  the  church;  he  never  rose  above 
being  rector  of  Croft. 

While  a  broad  churchman  the  Rev.  Mr.  Kirkman  was  very 
vehement  against  the  leaders  of  the  materialistic  philosophy. 
Two  years  after  Tyndall's  Belfast  address,  in  which  he  announced 
that  he  could  discern  in  matter  the  promise  and  potency  of 
every  form  of  life,  Kirkman  published  a  volume  entitled  Philoso- 
phy without  Assumptions,  in  which  he  criticises  in  very  vigorous 
style  the  materialistic  and  evolutional  philosophy  advocated  by 
Mill,  Spencer,  Tyndall,  and  Huxley.  In  ascribing  everything 
to  matter  and  its  powers  or  potencies  he  considers  that  they 
turn  philosophy  upside  down.  He  has,  he  writes,  first-hand 


130  TEN  BRITISH  MATHEMATICIANS 

knowledge  of  himself  as  a  continuous  person,  endowed  with 
will;  and  he  infers  that  there  are  will  forces  around;  but  he 
sees  no  evidence  of  the  existence  of  matter.  Matter  is  an 
assumption  and  forms  no  part  of  his  philosophy.  He  relies  on 
Boscovich's  theory  of  an  atom  as  simply  the  center  of  forces. 
Force  he  understands  from  his  knowledge  of  will,  but  any  other 
substance  he  does  not  understand.  The  obvious  difficulty  in 
this  philosophy  is  to  explain  the  belief  in  the  existence  of  other 
conscious  beings — other  will  forces.  Is  it  not  the  great  assump- 
tion which  everyone  is  obliged  to  make;  verified  by  experience, 
but  still  in  its  nature  an  assumption?  Kirkman  tries  to  get 
over  this  difficulty  by  means  of  a  syllogism,  the  major  premise 
of  which  he  has  to  manufacture,  and  which  he  presents  to  his 
reason  for  adoption  or  rejection.  How  can  a  universal  propo- 
sition be  easier  to  grasp  than  the  particular  case  included  in  it? 
If  the  mind  doubts  about  an  individual  case,  how  can  it  be  sure 
about  an  infinite  number  of  such  cases?  It  is  a  petitio  principii. 
As  a  critic  of  the  materialistic  philosophy  Kirkman  is  more 
successful.  He  criticises  Herbert  Spencer  on  free  will  as  follows : 
"  The  short  chapter  of  eight  pages  on  Will  cost  more  philosophi- 
cal toil  than  all  the  two  volumes  on  Psychology.  The  author 
gets  himself  in  a  heat,  he  runs  himself  into  a  corner,  and  brings 
himself  dangerously  to  bay.  Hear  him :  '  To  reduce  the  general 
question  to  its  simplest  form;  psychical  changes  either  conform 
to  law,  or  they  do  not.  If  they  do  not  conform  to  law,  this  work, 
in  common  with  all  other  works  on  the  subject,  is  sheer  non- 
sense; no  science  of  Psychology  is  possible.  If  they  do  conform 
to  law,  there  cannot  be  any  such  thing  as  free  will.'  Here  we 
see  the  horrible  alternative.  If  the  assertors  of  free  will  refuse 
to  commit  suicide,  they  must  endure  the  infinitely  greater  pang 
of  seeing  Mr.  Spencer  hurl  himself  and  his  books  into  that 
yawning  gulf,  a  sacrifice  long  devoted,  and  now  by  pitiless  Fate 
consigned,  to  the  abysmal  gods  of  nonsense.  Then  pitch  him 
down  say  I.  Shall  I  spare  him  who  tells  me  that  my  movements 
in  this  orbit  of  conscious  thought  and  responsibility  are  made 
under  'parallel  conditions'  with  those  of  yon  driven  moon? 
Shall  I  spare  him  who  has  juggled  me  out  of  my  Will,  my  noblest 


THOMAS  PENYNGTON   KIRKMAN  131 

attribute;  who  has  hocuspocused  me  out  of  my  subsisting 
personality;  and  then,  as  a  refinement  of  cruelty,  has  frightened 
me  out  of  the  rest  of  my  wits  by  forcing  me  to  this  terrific  alterna- 
tive that  either  the  testimony  of  this  Being,  this  Reason  and  this 
Conscience  is  one  ever-thundering  lie,  or  else  he,  even  he,  has 
talked  nonsense?  He  has  talked  nonsense,  I  say  it  because  I 
have  proved  it.  And  every  man  must  of  course  talk  nonsense 
who  begins  his  philosophy  with  abstracts  in  the  clouds  instead 
of  building  on  the  witness  of  his  own  self-consciousness.  '  If 
they  do  conform  to  law,'  says  Spencer, '  there  cannot  be  any  such 
thing  as  free  will.'  The  force  of  this  seems  to  depend  on  his 
knowledge  of  '  law.'  When  I  ask,  What  does  this  writer  know 
of  law — definite  working  law  in  the  Cosmos? — the  only  answer 
I  can  get  is — Nothing,  except  a  very  little  which  he  has  picked 
up,  often  malappropriately,  as  we  have  seen,  among  the  mathe- 
maticians. When  I  ask — What  does  he  know  about  law? — there 
is  neither  beginning  nor  end  to  the  reply.  I  am  advised  to  read 
his  books  about  law,  and  to  master  the  differentiations  and  inte- 
grations of  the  coherences,  the  correlations,  the  uniformities, 
and  universalities  which  he  has  established  in  the  abstract  over 
all  space  and  all  time  by  his  vast  experience  and  miraculous 
penetration.  I  have  tried  to  do  this,  and  have  found  all  pretty 
satisfactory,  except  the  lack  of  one  thing — something  like  proof 
of  his  competence  to  decide  all  that  scientifically.  When  I 
persist  in  my  demand  for  such  proof,  it  turns  out  at  last — that  he 
knows  by  heart  the  whole  Hymn  Book,  the  Litanies,  the  Missal, 
and  the  Decretals  of  the  Must-be-ite  religion!  '  Conform  to 
law.'  Shall  I  tell  you  what  he  means  by  that?  Exactly  ninety- 
nine  hundred ths  of  his  meaning  under  the  word  law  is  must  be." 
Kirkman  points  out  that  the  kind  of  proof  offered  by  these 
philosophers  is  a  bold  assertion  of  must-be-so.  For  instance 
he  mentions  Spencer's  evolution  of  consciousness  out  of  the 
unconscious:  "That  an  effectual  adjustment  may  be  made 
they  (the  separate  impressions  or  constituent  changes  of  a  com- 
plex correspondence  to  be  coordinated)  must  be  brought  into 
relation  with  each  other.  But  this  implies  some  center  of  com- 
munication common  to  them  all,  through  which  they  severally 


132  TEN  BRITISH  MATHEMATICIANS 

pass;  and  as  they  cannot  pass  through  it  simultaneously,  they 
must  pass  through  it  in  succession.  So  that  as  the  external 
phenomena  responded  to  become  greater  in  number  and  more 
complicated  in  kind,  the  variety  and  rapidity  of  the  changes 
to  which  this  common  center  of  communication  is  subject  must 
increase,  there  must  result  an  unbroken  series  of  those  changes, 
there  must  arise  a  consciousness." 

The  paraphrase  which  Kirkman  gave  .of  Spencer's  definition 
of  Evolution  commended  itself  to  such  great  minds  as  Tait  and 
Clerk-Maxwell.  Spencer's  definition  is:  "  Evolution  is  a  change 
from  an  indefinite  incoherent  homogeneity  to  a  definite  coherent 
heterogeneity,  through  continuous  differentiations  and  integra- 
tions." Kirkman's  paraphrase  is  "  Evolution  is  a  change  from 
a  nohowish  untalkaboutable  all-likeness,  to  a  somehowish  and 
in-general-talkaboutable  not-all-likeness,  by  continuous  some- 
thingelseifications  and  sticktogetherations."  The  tone  of  Kirk- 
man's book  is  distinctly  polemical  and  full  of  sarcasm.  He 
unfortunately  wrote  as  a  theologian  rather  than  as  a  mathe- 
matician. The  writers  criticised  did  not  reply,  although  they 
felt  the  edge  of  his  sarcasm;  and  they  acted  wisely,  for  they 
could  not  successfully  debate  any  subject  involving  exact  science 
against  one  of  the  most  penetrating  mathematicians  of  the 
nineteenth  century. 

We  have  seen  that  Hamilton  appreciated  Kirkman's  genius; 
so  did  Cayley,  De  Morgan,  Clerk-Maxwell,  Tait.  One  of  Tait's 
most  elaborate  researches  was  the  enumeration  and  construc- 
tion of  the  knots  which  can  be  formed  in  an  endless  cord — a 
subject  which  he  was  induced  to  take  up  on  account  of  its  bear- 
ing on  the  vortex  theory  of  atoms.  If  the  atoms  are  vortex 
filaments  their  differences  in  kind,  giving  rise  to  differences  in 
the  spectra  of  the  elements,  must  depend  on  a  greater  or  less 
complexity  in  the  form  of  the  closed  filament,  and  this  difference 
would  depend  on  the  knottiness  of  the  filament.  Hence  the 
main  question  was  "  How  many  different  forms  of  knots  are 
there  with  any  given  small  number  of  crossings?"  Tait  made 
the  investigation  for  three,  four,  five,  six,  seven,  eight  cross- 
ings. Kirkman's  investigations  on  the  polyedra  were  much 


THOMAS   PENYNGTON   KIRKMAN  133 

allied.  He  took  up  the  problem  and,  with  some  assistance 
from  Tait,  solved  it  not  only  for  nine  but  for  ten  crossings.  An 
investigation  by  C.  N.  Little,  a  graduate  of  Yale  University, 
has  confirmed  Kirkman's  results. 

Through  Professor  Tait  I  was  introduced  to  Rev.  Mr.  Kirk- 
man;  and  we  discussed  the  mathematical  analysis  of  relation- 
ships, formal  logic,  and  other  subjects.  After  I  had  gone  to  the 
University  of  Texas,  Kirkman  sent  me  through  Tait  the  follow- 
ing question  which  he  said  was  current  in  society :  "  Two  boys, 
Smith  and  Jones,  of  the  same  age,  are  each  the  nephew  of  the 
other;  how  many  legal  solutions?  "  I  set  the  analysis  to  work, 
wrote  out  the  solutions,  and  the  paper  is  printed  in  the  Proceed- 
ings of  the  Royal  Society  of  Edinburgh.  There  are  four  solu- 
tions, rovided  Smith  and  Jones  are  taken  to  be  mere  arbitrary, 
names;  if  the  convention  about  surnames  holds  there  are  only 
two  legal  solutions.  On  seeing  my  paper  Kirkman  sent  the 
question  to  the  Educational  Times  in  the  following  improved 

form: 

Baby  Tom  of  baby  Hugh 

The  nephew  is  and  uncle  too; 

In  how  many  ways  can  this  be  true? 

Thomas  Penyngton  Kirkman  died  on  February  3,  1895, 
having  very  nearly  reached  the  age  of  89  years.  I  have  found 
only  one  printed  notice  of  his  career,  but  all  his  writings  are 
mentioned  in  the  new  German  Encyclopaedia  of  Mathematics. 
He  was  an  honorary  member  of  the  Literary  and  Philosophical 
Societies  of  Manchester  and  of  Liverpool,  a  Fellow  of  the  Royal 
Society,  and  a  foreign  member  of  the  Dutch  Society  of  Sciences 
at  Haarlem.  I  may  close  by  a  quotation  from  one  of  his  letters: 
"  What  I  have  done  in  helping  busy  Tait  in  knots  is,  like  the 
much  more  difficult  and  extensive  things  I  have  done  in  polyedra 
or  groups,  not  at  likely  to  be  talked  about  intelligently  by  people 
so  long  as  I  live.  But  it  is  a  faint  pleasure  to  think  it  will  one 
day  win  a  little  praise." 


ISAAC   TODHUNTER* 

(1820-1884) 

ISAAC  TODHUNTER  was  born  at  Rye,  Sussex,  23  Nov.,  1820. 
He  was  the  second  son  of  George  Todhunter,  Congregationalist 
minister  of  the  place,  and  of  Mary  his  wife,  whose  maiden  name 
was  Hume,  a  Scottish  surname.  The  minister  died  of  con- 
sumption when  Isaac  was  six  years  old,  and  left  his  family, 
consisting  of  wife  and  four  boys,  in  narrow  circumstances.  The 
widow,  who  was  a  woman  of  strength,  physically  and  mentally, 
moved  to  the  larger  town  of  Hastings  in  the  same  county,  and 
opened  a  school  for  girls.  After  some  years  Isaac  was  sent  to 
a  boys'  school  in  the  same  town  kept  by  Robert  Carr,  and  sub- 
sequently to  one  newly  opened  by  a  Mr.  Austin  from  London; 
for  some  years  he  had  been  unusually  backward  in  his  studies, 
but  under  this  new  teacher  he  made  rapid  progress,  and  his 
career  was  then  largely  determined. 

After  his  school  days  were  over,  he  became  an  usher  or 
assistant  master  with  Mr.  Austin  in  a  school  at  Peckham;  and 
contrived  to  attend  at  the  same  time  the  evening  classes  at 
University  College,  London.  There  he  came  under  the  great 
educating  influence  of  De  Morgan,  for  whom  in  after  years  he 
always  expressed  an  unbounded  admiration;  to  De  Morgan 
"  he  owed  that  interest  in  the  history  and  bibliography  of 
science,  in  moral  philosophy  and  logic  which  determined  the 
course  of  his  riper  studies."  In  1839  he  passed  the  matriculation 
examination  of  the  University  of  London,  then  a  merely  examin- 
ing body,  winning  the  exhibition  for  mathematics  (£30  for  two 
years);  in  1842  he  passed  the  B.A.  examination  carrying  off 
a  mathematical  scholarship  (of  £50  for  three  years) ;  and  in  1844 
obtained  the  degree  of  Master  of  Arts  with  the  gold  medal 

*  This  Lecture  w^s  delivered.  April  13,  1904. — EDITORS. 
134 


ISAAC  TODHUNTER  135 

awarded  to  the  candidate  who  gained  the  greatest  distinction 
in  that  examination. 

Sylvester  was  then  professor  of  natural  philosophy  in  Uni- 
versity College,  and  Todhunter  studied  under  him.  The 
writings  of  Sir  John  Herschel  also  had  an  influence;  for  Tod- 
hunter  wrote  as  follows  (Conflict  of  Studies,  p.  66) :  "  Let  me 
at  the  outset  record  my  opinion  of  mathematics;  I  cannot  do 
this  better  than  by  adopting  the  words  of  Sir  J.  Herschel, 
to  the  influence  of  which  I  gratefully  attribute  the  direction  of 
my  own  early  studies.  He  says  of  Astronomy,  '  Admission  to 
its  sanctuary  can  only  be  gained  hy  one  means, — sound  and 
sufficient  knowledge  of  mathematics,  the  great  instrument  of 
all  exact  inquiry,  without  which  no  man  can  ever  make  such 
advances  in  this  or  any  other  of  the  higher  departments  of 
science  as  can  entitle  him  to  form  an  independent  opinion  on 
any  subject  of  discussion  within  their  range.' ' 

When  Todhunter  graduated  as  M.A.  he  was  24  years  of  age. 
Sylvester  had  gone  to  Virginia,  but  De  Morgan  remained.  The 
latter  advised  him  to  go  through  the  regular  course  at  Cambridge; 
his  name  was  now  entered  at  St.  John's  College.  Being  some- 
what older,  and  much  more  brilliant  than  the  honor  men  of 
his  year,  he  was  able  to  devote  a  great  part  of  his  attention  to 
studies  beyond  those  prescribed.  Among  other  subjects  he 
took  up  Mathematical  Electricity.  In  1848  he  took  his  B.A. 
degree  as  senior  wrangler,  and  also  won  the  first  Smith's  prize. 

While  an  undergraduate  Todhunter  lived  a  very  secluded 
life.  He  contributed  along  with  his  brothers  to  the  support 
of  their  mother,  and  he  had  neither  money  nor  time  to  spend 
on  entertainments.  The  following  legend  was  applied  to  him, 
if  not  recorded  of  him:  "Once  on  a  time,  a  senior  wrangler 
gave  a  wine  party  to  celebrate  his  triumph.  Six  guests  took 
their  seats  round  the  table.  Turning  the  key  in  the  door,  he 
placed  one  bottle  of  wine  on  the  table  asseverating  with  unction, 
'  None  of  you  will  leave  this  room  while  a  single  drop  remains.'  " 

At  the  University  of  Cambridge  there  is  a  foundation  which 
provides  for  what  is  called  the  Burney  prize.  According  to  the 
regulations  the  prize  is  to  be  awarded  to  a  graduate  of  the 


136  TEN   BRITISH   MATHEMATICIANS 

University  who  is  not  of  more  than  three  years'  standing  from 
admission  to  his  degree  and  who  shall  produce  the  best  English 
essay  "  On  some  moral  or  metaphysical  subject,  or  on  the 
existence,  nature  and  attributes  of  God,  or  on  the  truth  and 
evidence  of  the  Christian  religion."  Todhunter  in  the  course 
of  his  first  postgraduate  year  submitted  an  essay  on  the  thesis 
that  "  The  doctrine  of  a  divine  providence  is  inseparable  from 
the  belief  in  the  existence  of  an  absolutely  perfect  Creator." 
This  essay  received  the  prize,  and  was  printed  in  1849. 

Todhunter  now  proceeded  to  the  degree  of  M.A.,  and  unlike 
his  mathematical  instructors  in  University  College,  De  Morgan 
and  Sylvester,  he  did  not  parade  his  non-conformist  principles, 
but  submitted  to  the  regulations  with  as  good  grace  as  possible. 
He  was  elected  a  fellow  of  his  college,  but  not  immediately, 
probably  on  account  of  his  being  a  non-conformist,  and  appointed 
lecturer  on  mathematics  therein;  he  also  engaged  for  some  time 
in  work  as  a  private  tutor,  having  for  one  of  his  pupils  P.  G. 
Tait,  and  I  believe  E.  J.  Routh  also. 

For  a  space  of  15  years  he  remained  a  fellow  of  St.  John's 
College,  residing  in  it,  and  taking  part  in  the  instruction.  He 
was  very  successful  as  a  lecturer,  and  it  was  not  long  before  he 
began  to  publish  textbooks  on  the  subjects  of  his  lectures.  In 
1853  he  published  a  textbook  on  Analytical  Statics;  in  1855 
one  on  Plane  Coordinate  Geometry;  and  in  1858  Examples  of 
Analytical  Geometry  of  Three  Dimensions.  His  success  in  these 
subjects  induced  him  to  prepare  manuals  on  elementary  mathe- 
matics; his  Algebra  appeared  in  1858,  his  Trigonometry  in  1859, 
his  Theory  of  Equations  in  1861,  and  his  Euclid  in  1862.  Some 
of  his  textbooks  passed  through  many  editions  and  have  been 
widely  used  in  Great  Britain  and  North  America.  Latterly 
he  was  appointed  principal  mathematical  lecturer  in  his  college, 
and  he  chose  to  drill  the  freshmen  in  Euclid  and  other  elemen- 
tary mathematics. 

Within  these  years  he  also  labored  at  some  works  of  a  more 
strictly  scientific  character.  Professor  Woodhouse  (who  was 
the  forerunner  of  the  Analytical  Society)  had  written  a  history 
of  the  calculus  of  variations,  ending  with  the  eighteenth  century; 


ISAAC  TODHUNTER  137 

this  work  was  much  admired  for  its  usefulness  by  Todhunter, 
and  as  he  felt  a  decided  taste  for  the  history  of  mathematics, 
he  formed  and  carried  out  the  project  of  continuing  the  history 
of  that  calculus  during  the  nineteenth  century.  It  was  the 
first  of  the  great  historical  works  which  has  given  Todhunter 
his  high  place  among  the  mathematicians  of  the  nineteenth 
century.  This  history  was  published  in  1861;  in  1862  he  was 
elected  a  Fellow  of  the  Royal  Society  of  London.  In  1863  he 
was  a  candidate  for  the  Sadlerian  professorship  of  Mathe- 
matics, to  which  Cayley  was  appointed.  Todhunter  was  not 
a  mere  mathematical  specialist.  He  was  an  excellent  linguist; 
besides  being  a  sound  Latin  and  Greek  scholar,  he  was  familiar 
with  French,  German,  Spanish,  Italian  and  also  Russian,  Hebrew 
and  Sanskrit.  He  was  likewise  well  versed  in  philosophy,  and 
for  the  two  years  1863-5  acted  as  an  Examiner  for  the  Moral 
Science  Tripos,  of  which  the  chief  founders  were  himself  and 
Whewell. 

By  1864  the  financial  success  of  his  books  was  such  that  he 
was  able  to  marry,  a  step  which  involved  the  resigning  of  his 
fellowship.  His  wife  was  a  daughter  of  Captain  George  Davies 
of  the  Royal  Navy,  afterwards  Admiral  Davies. 

As  a  fellow  and  tutor  of  St.  John's  College  he  had  lived  a 
very  secluded  life.  His  relatives  and  friends  thought  he  was  a 
confirmed  bachelor.  He  had  sometimes  hinted  that  the  grapes 
were  sour.  For  art  he  had  little  eye;  for  music  no  ear.  "  He 
used  to  say  he  knew  two  tunes;  one  was  '  God  save  the  Queen,' 
the  other  wasn't.  The  former  he  recognized  by  the  people 
standing  up."  As  owls  shun  the  broad  daylight  he  had  shunned 
the  glare  of  parlors.  It  was  therefore  a  surprise  to  his  friends 
and  relatives  when  they  were  invited  to  his  marriage  in  1864. 
Prof.  Mayor  records  that  Todhunter  wrote  to  his  fiancee,  "  You 
will  not  forget,  I  am  sure,  that  I  have  always  been  a  student, 
and  always  shall  be;  but  books  shall  not  come  into  even  distant 
rivalry  with  you,"  and  Prof.  Mayor  insinuated  that  thus  fore- 
armed, he  calmly  introduced  to  the  inner  circle  of  their  honey- 
moon Hamilton  on  Quaternions. 

It  was  now  (1865)  that  the  London  Mathematical  Society 


138  TEN  BRITISH   MATHEMATICIANS 

was  organized  under  the  guidance  of  DC  Morgan,  and  Todhunter 
became  a  member  in  the  first  year  of  its  existence.  The  same 
year  he  discharged  the  very  onerous  duties  of  examiner  for  the 
mathematical  tripos — a  task  requiring  so  much  labor  and 
involving  so  much  interference  with  his  work  as  an  author 
that  he  never  accepted  it  again.  Now  (1865)  appeared  his 
History  of  the  Mathematical  Theory  of  Probability,  and  the  same 
year  he  was  able  to  edit  a  new  edition  of  Boole's  Treatise  on 
Differential  Equations,  the  author  having  succumbed  to  an 
untimely  death.  Todhunter  certainly  had  a  high  appreciation 
of  Boole,  which  he  shared  in  common  with  De  Morgan.  The 
work  involved  in  editing  the  successive  editions  of  his  elementary 
books  was  great;  he  did  not  proceed  to  stereotype  until  many 
independent  editions  gave  ample  opportunity  to  correct  all 
errors  and  misprints.  He  now  added  two  more  textbooks; 
Mechanics  in  1867  and  Mensuration  in  1869. 

About  1847  ^e  members  of  St.  John's  College  founded  a 
prize  in  honor  of  their  distinguished  fellow,  J.  C.  Adams.  It 
is  awarded  every  two  years,  and  is  in  value  about  £225.  In 
1869  the  subject  proposed  was  "  A  determination  of  the  circum- 
stances under  which  Discontinuity  of  any  kind  presents  itself 
in  the  solution  of  a  problem  of  maximum  or  minimum  in  the 
Calculus  of  Variations."  There  had  been  a  controversy  a  few 
years  previous  on  this  subject  in  the  pages  of  Philosophical 
Magazine  and  Todhunter  had  there  advocated  his  view  of  the 
matter.  "  This  view  is  found  in  the  opening  sentences  of  his 
essay:  'We  shall  find  that,  generally  speaking,  discontinuity  is 
introduced,  by  virtue  of  some  restriction  which  we  impose,  either 
explicitly  or  implicitly  in  the  statement  of  the  problems  which 
we  propose  to  solve.'  This  thesis  he  supported  by  considering 
in  turn  the  usual  applications  of  the  calculus,  and  pointing  out 
where  he  considers  the  discontinuities  which  occur  have  been 
introduced  into  the  conditions  of  the  problem.  This  he  success- 
fully proves  in  many  instances.  In  some  cases,  the  want  of  a 
distinct  test  of  what  discontinuity  is  somewhat  obscures  the 
argument."  To  his  essay  the  prize  was  awarded ;  it  is  published 
under  the  title  "  Researches  in  the  Calculus  of  Variations  " — 


ISAAC  TODHUNTER  139 

an  entirely  different  work  from  his  History  of  the  Calculus  of 
Variations. 

In  1873  ne  published  his  History  of  the  Mathematical 
Theories  of  Attraction.  It  consists  of  two  volumes  of  nearly  1000 
pages  altogether  and  is  probably  the  most  elaborate  of  his 
histories.  In  the  same  year  (1873)  he  published  in  book  form 
his  views  on  some  of  the  educational  questions  of  the  day, 
under  the  title  of  The  Corflict  of  Studies,  and  other  essays  on 
subjects  connected  with  education.  The  collection  contains 
six  essays;  they  were  originally  written  with  the  view  of  suc- 
cessive publication  in  some  magazine,  but  in  fact  they  were 
published  only  in  book  form.  In  the  first  essay,  that  on  the 
Conflict  of  Studies — Todhunter  gave  his  opinion  of  the  edu- 
cative value  in  high  schools  and  colleges  of  the  different  kinds 
of  study  then  commonly  advocated  in  opposition  to  or  in 
addition  to  the  old  subjects  of  classics  and  mathematics.  He 
considered  that  the  Experimental  Sciences  were  little  suitable, 
and  that  for  a  very  English  reason,  because  they  could  not  be 
examined  on  adequately.  He  says: 

"  Experimental  Science  viewed  in  connection  with  educa- 
tion, rejoices  in  a  name  which  is  unfairly  expressive.  A  real 
experiment  is  a  very  valuable  product  of  the  mind,  requiring 
great  knowledge  to  invent  it  and  great  ingenuity  to  carry  it 
out.  When  Perrier  ascended  the  Puy  de  Dome  with  a  barom- 
eter in  order  to  test  the  influence  of  change  of  level  on  the 
height  of  the  column  of  mercury,  he  performed  an  experiment, 
the  suggestion  of  which  was  worthy  of  the  genius  of  Pascal 
and  Descartes.  But  when  a  modern  traveller  ascends  Mont 
Blanc,  and  directs  one  of  his  guides  to  carry  a  barometer,  he 
cannot  be  said  to  perform  an  experiment  in  any  very  exact  or 
very  meritorious  sense  of  the  word.  It  is  a  repetition  of  an 
observation  made  thousands  of  times  before,  and  we  can  never 
recover  any  of  the  interest  which  belonged  to  the  first  trial, 
unless  indeed,  without  having  ever  heard  of  it,  we  succeeded 
in  reconstructing  the  process  of  ourselves.  In  fact,  almost 
always  he  who  first  plucks  an  experimental  flower  thus  ap- 
propriates and  destroys  its  fragrance  and  its  beauty." 


140  TEN   BRITISH   MATHEMATICIANS 

At  the  time  when  Todhimter  was  writing  the  above,  the 
Cavendish  Laboratory  for  Experimental  Physics  was  just  being 
built  at  Cambridge,  and  Clerk-Maxwell  had  just  been  appointed 
the  professor  of  the  new  study;  from  Todhunter's  utterance 
we  can  see  the  state  of  affairs  then  prevailing.  Consider  the 
corresponding  experiment  of  Torricelli,  which  can  be  performed 
inside  a  classroom;  to  every  fresh  student  the  experiment 
retains  its  fragrance;  the  sight  of  it,  and  more  especially  the 
performance  of  it  imparts  a  kind  of  knowledge  which  cannot  be 
got  from  description  or  testimony;  it  imparts  accurate  concep- 
tions and  is  a  necessary  preparative  for  making  a  new  and 
original  experiment.  To  Todhunter  it  may  be  replied  that  the 
flowers  of  Euclid's  Elements  were  plucked  at  least  2000  years 
ago,  yet,  he  must  admit,  they  still  possess,  to  the  fresh  student 
of  mathematics,  even  although  he  becomes  acquainted  with 
them  through  a  textbook,  both  fragrance  and  beauty." 

Todhunter  went  on  to  write  another  passage  which  roused 
the  ire  of  Professor  Tait.  "  To  take  another  example.  We 
assert  that  if  the  resistance  of  the  air  be  withdrawn  a  sovereing 
and  a  feather  will  fall  through  equal  spaces  in  equal  times. 
Very  great  credit  is  due  to  the  person  who  first  imagined  the 
well-known  experiment  to  illustrate  this;  but  it  is  not  obvious 
what  is  the  special  benefit  now  gained  by  seeing  a  lecturer  repeat 
the  process.  It  may  be  said  that  a  boy  takes  more  interest  in 
the  matter  by  seeing  for  himself,  or  by  performing  for  himself, 
that  is,  by  working  the  handle  of  the  air-pump;  this  we  admit, 
while  we  continue  to  doubt  the  educational  value  of  the  trans- 
action. The  boy  would  also  probably  take  much  more  interest 
in  football  than  in  Latin  grammar;  but  the  measure  of  his 
interest  is  not  identical  with  that  of  the  importance  of  the 
subjects.  It  may  be  said  that  the  fact  makes  a  stronger  impres- 
sion on  the  boy  through  the  medium  of  his  sight,  that  he  believes 
it  the  more  confidently.  I  say  that  this  ought  not  to  be  the  case. 
If  he  does  not  believe  the  statements  of  his  tutor — probably 
a  clergyman  of  mature  knowledge,  recognized  ability  and  blame- 
less character — his  suspicion  is  irrational,  and  manifests  a  want 
of  the  power  of  appreciating  evidence,  a  want  fatal  to  his  sue- 


ISAAC   TODHUNTER  141 

cess  in  that  branch  of  science  which  he  is  supposed  to  be  cul- 
tivating." 

Clear  physical  conceptions  cannot  be  got  by  tradition,  even 
from  a  clergyman  of  blameless  charater;  they  are  best  got 
directly  from  Nature,  and  this  is  recognized  by  the  modern 
laboratory  instruction  in  physics.  Todhunter  would  reduce 
science  to  a  matter  of  authority;  and  indeed  his  mathematical 
manuals  are  not  free  from  that  fault.  He  deals  with  the  charac- 
teristic difficulties  of  algebra  by  authority  rather  than  by  sci- 
entific explanation.  Todhunter  goes  on  to  say:  "  Some  con- 
siderable drawback  must  be  made  from  the  educational  value 
of  experiments,  so  called,  on  account  of  their  failure.  Many 
persons  must  have  been  present  at  the  exhibitions  of  skilled 
performers,  and  have  witnessed  an  uninterrupted  series  of 
ignominious  reverses, — they  have  probably  longed  to  imitate  the 
cautious  student  who  watched  an  eminent  astronomer  baffled 
by  Foucault's  experiment  for  proving  the  rotation  of  the  Earth; 
as  the  pendulum  would  move  the  wrong  way  the  student  retired, 
saying  that  he  wished  to  retain  his  faith  in  the  elements  of 
astronomy." 

It  is  not  unlikely  that  the  series  of  ignominious  reverses 
Todhunter  had  in  his  view  were  what  he  had  seen  in  the  physics 
classroom  of  University  College  when  the  manipulation  was 
in  the  hands  of  a  pure  mathematician — Prof.  Sylvester.  At 
the  University  of  Texas  there  is  a  fine  clear  space  about  60 
feet  high  inside  the  building,  very  suitable  for  Foucault's  experi- 
ment. I  fixed  up  a  pendulum,  using  a  very  heavy  ball,  and 
the  turning  of  the  Earth  could  be  seen  in  two  successive  oscilla- 
tions. The  experiment,  although  only  a  repetition  according  to 
Todhunter,  was  a  live  and  inspiring  lesson  to  all  who  saw  it, 
whether  they  came  with  previous  knowledge  about  it  or  no. 
The  repetition  of  any  such  great  experiment  has  an  educative 
value  of  which  Todhunter  had  no  conception. 

Another  subject  which  Todhunter  discussed  in  these  essays 
is  the  suitability  of  Euclid's  Elements  for  use  as  the  elementary 
textbook  of  Geometry.  His  experience  as  a  college  tutor  for 
25  years;  his  numerous  engagements  as  an  examiner  in  mathe- 


142  TEN   BRITISH  MATHEMATICIANS 

matics;  his  correspondence  with  teachers  in  the  large  schools 
gave  weight  to  the  opinion  which  he  expressed.  The  question 
was  raised  by  the  first  report  of  the  Association  for  the  Improve- 
ment of  Geometrical  Teaching;  and  the  points  which  Todhunter 
made  were  afterwards  taken  up  and  presented  in  his  own  unique 
style  by  Lewis  Carroll  in  "  Euclid  and  his  modern  rivals."  Up 
to  that  time  Euclid's  manual  was,  and  in  a  very  large  measure 
still  is,  the  authorized  introduction  to  geometry;  it  is  not  as  in 
this  country  where  there  is  perfect  liberty  as  to  the  books  and 
methods  to  be  employed.  The  great  difficulty  in  the  way  of 
liberty  in  geometrical  teaching  is  the  universal  tyranny  of  com- 
petitive examinations.  Great  Britain  is  an  examination-ridden 
country.  Todhunter  referred  to  one  of  the  most  distinguished 
professors  of  Mathematics  in  England;  one  whose  pupils  had 
likewise  gained  a  high  reputation  as  investigators  and  teachers; 
his  "  venerated  master  and  friend,"  Prof/  De  Morgan;  and 
pointed  out  that  he  recommended  the  study  of  Euclid  with  all 
the  authority  of  his  great  attainments  and  experience. 

Another  argument  used  by  Todhunter  was  as  follows:  In 
America  there  are  the  conditions  which  the  Association  desires; 
there  is,  for  example,  a  textbook  which  defines  parallel  lines  as 
those  which  have  the  same  direction.  Could  the  American  mathe- 
maticians of  that  day  compare  with  those  of  England?  He 
answered  no. 

While  Todhunter  could  point  to  one  master— De  Morgan — 
as  in  his  favor,  he  was  obliged  to  quote  another  master — Syl- 
vester— as  opposed.  In  his  presidential  address  before  section  A 
of  the  British  Association  at  Exeter  in  1869,  Sylvester  had  said: 
"  I  should  rejoice  to  see  ...  Euclid  honorably  shelved  or  buried 
'  deeper  than  did  ever  plummet  sound  '  out  of  the  schoolboy's 
reach;  morphology  introduced  into  the  elements  of  algebra; 
projection,  correlation,  and  motion  accepted  as  aids  to  geometry; 
the  mind  of  the  student  quickened  and  elevated  and  his  faith 
awakened  by  early  initiation  into  the  ruling  ideas  of  polarity, 
continuity,  infinity,  antl  familiarization  with  the  doctrine  of 
the  imaginary  and  inconceivable."  Todhunter  replied:  "  What- 
ever may  have  produced  the  dislike  to  Euclid  in  the  illustrious 


ISAAC  TODIIUNTER  143 

mathematician  whose  words  I  have  quoted,  there  is  no  ground 
for  supposing  that  he  would  have  been  better  pleased  with  the 
substitutes  which  are  now  offered  and  recommended  in  its  place. 
But  the  remark  which  is  naturally  suggested  by  the  passage : 
is  that  nothing  prevents  an  enthusiastic  teacher  from  carrying 
his  pupils  to  any  height  he  pleases  in  geometry,  even  if  he  starts 
with  the  use  of  Euclid." 

Todhunter  also  replied  to  the  adverse  opinion,  delivered  by 
some  professor  (doubtless  Tait)  in  an  address  at  Edinburgh 
which  was  as  follows:  "  From  the  majority  of  the  papers  in  our 
few  mathematical  journals,  one  would  almost  be  led  to  fancy 
that  British  mathematicians  have  too  much  pride  to  use  a  simple 
method,  while  an  unnecessarily  complex  one  can  be  had.  No 
more  telling  example  of  this  could  be  wished  for  than  the  insane 
delusion  under  which  they  permit  '  Euclid  '  to  be  employed  in 
our  elementary  teaching.  They  seem  voluntarily  to  weight 
alike  themselves  and  their  pupils  for  the  race."  To  which 
Todhunter  replied:  "The  British  mathematical  journals  with 
the  titles  of  which  I  am  acquainted  are  the  Quarterly  Journal  of 
Mathematics,  the  Mathematical  Messenger,  and  the  Philosoph- 
ical Magazine;  to  which  may  be  added  the  Proceedings  of  the 
Royal  Society  and  the  Monthly  Notices  of  the  Astronomical 
Society.  I  should  have  thought  it  would  have  been  an  adequate 
employment,  for  a  person  engaged  in  teaching,  to  read  and  master 
these  periodicals  regularly;  but  that  a  single  mathematician 
should  be  able  to  improve  more  than  half  the  matter  which  is 
thus  presented  to  him  fills  me  with  amazement.  I  take  down 
some  of  these  volumes,  and  turning  over  the  pages  I  find  article 
after  article  by  Profs.  Cayley,  Salmon  and  Sylvester,  not  to 
mention  many  other  highly  distinguished  names.  The  idea  of 
amending  the  elaborate  essays  of  these  eminent  mathematicians 
seems  to  me  something  like  the  audacity  recorded  in  poetry 
with  which  a  superhuman  hero  climbs  to  the  summit  of 
the  Indian  Olympus  and  overturns  the  thrones  of  Vishnu, 
Brahma  and  Siva.  While  we  may  regret  that  such  ability 
should  be  exerted  on  the  revolutionary  side  of  the  question, 
here  is  at  least  one  mournful  satisfaction:  the  weapon  with 


144  TEN  BRITISH  MATHEMATICIANS 

which  Euclid  is  assailed  was  forged  by  Euclid  himself.  The 
justly  celebrated  professor,  from  whose  address  the  quotation 
is  taken,  was  himself  trained  by  those  exercises  which  he  now 
considers  worthless;  twenty  years  ago  his  solutions  of  mathe- 
matical problems  were  rich  with  the  fragrance  of  the  Greek 
geometry.  I  venture  to  predict  that  we  shall  have  to  wait 
some  time  before  a  pupil  will  issue  from  the  reformed  school, 
who  singlehanded  will  be  able  to  challenge  more  than  half  the 
mathematicians  of  England."  Professor  Tait,  in  what  he  said, 
had,  doubtless,  reference  to  the  avoidance  of  the  use  of  the 
Quaternion  method  by  his  contemporaries  in  mathematics. 

More  than  half  of  the  Essays  is  taken  up  with  questions 
connected  with  competitive  examinations.  Todhunter  explains 
the  influence  of  Cambridge  in  this  matter:  "  Ours  is  an  age 
of  examination;  and  the  University  of  Cambridge  may  claim 
the  merit  of  originating  this  characteristic  of  the  period.  When 
we  hear,  as  we  often  do,  that  the  Universities  are  effete  bodies 
which  have  lost  their  influence  on  the  national  character,  we  may 
point  with  real  or  affected  triumph  to  the  spread  of  examinations 
as  a  decisive  proof  that  the  humiliating  assertion  is  not  absolutely 
true.  Although  there  must  have  been  in  schools  and  elsewhere 
processes  resembling  examinations  before  those  of  Cambridge 
had  become  widely  famous,  yet  there  can  be  little  chance  of 
error  in  regarding  our  mathematical  tripos  as  the  model  for 
rigor,  justice  and  importance,  of  a  long  succession  of  insti- 
tutions of  a  similar  kind  which  have  since  been  constructed." 
Todhunter  makes  the  damaging  admission  that  "  We  cannot 
by  our  examinations,  create  learning  or  genius;  it  is  uncertain 
whether  we  can  infallibly  discover  them;  what  we  detect  is  simply 
the  examination-passing  power." 

In  England  education  is  for  the  most  part  directed  to  train- 
ing pupils  for  examination.  One  direct  consequence  is  that 
the  memory  is  cultivated  at  the  expense  of  the  understanding; 
knowledge  instead  of  being  assimilated  is  crammed  for  the  time 
being,  and  lost  as  soon  as  the  examination  is  over.  Instead  of 
a  rational  study  of  the  principles  of  mathematics,  attention  is 
directed  to  problem-making, — to  solving  ten-minute  conun- 


ISAAC   TODHUNTER  145 

drums.  Textbooks  are  written  with  the  view  not  of  teaching 
the  subject  in  the  most  scientific  manner,  but  of  passing  certain 
specified  examinations.  I  have  seen  such  a  textbook  on  trigo- 
nometry where  all  the  important  theorems  which  required  the 
genius  of  Gregory  and  others  to  discover,  are  put  down  as  so 
many  definitions.  Nominal  knowledge,  not  real,  is  the  kind 
that  suits  examinations. 

Todhunter  possessed  a  considerable  sense  of  humour.  We 
see  this  in  his  Essays;  among  other  stories  he  tells  the  following: 
A  youth  who  was  quite  unable  to  satisfy  his  examiners  as  to  a 
problem,  endeavored  to  mollify  them,  as  he  said,  "  by  writing 
out  book  work  bordering  on  the  problem."  Another  youth 
who  was  rejected  said  "  if  there  had  been  fairer  examiners  and 
better  papers  I  should  have  passed;  I  knew  many  things  which 
were  not  set."  Again:  "A  visitor  to  Cambridge  put  himself 
under  the  care  of  one  of  the  self-constituted  guides  who  obtrude 
their  services.  Members  of  the  various  ranks  of  the  academical 
state  were  pointed  out  to  the  stranger — heads  of  colleges,  pro- 
fessors and  ordinary  fellows;  and  some  attempt  was  made  to 
describe  the  nature  of  the  functions  discharged  by  the  heads 
and  professors.  But  an  inquiry  as  to  the  duties  of  fellows  pro- 
duced and  reproduced  only  the  answer,  '  Them's  fellows  I  say.' 
The  guide  had  not  been  able  to  attach  the  notion  of  even  the 
pretense  of  duty  to  a  fellowship." 

In  1874  Todhunter  was  elected  an  honorary  fellow  of  his 
college,  an  honor  which  he  prized  very  highly.  Later  on  he 
was  chosen  as  an  elector  to  three  of  the  University  professor- 
ships— Moral  Philosophy,  Astronomy,  Mental  Philosophy  and 
Logic.  "  When  the  University  of  Cambridge  established  its 
new  degree  of  Doctor  of  Science,  restricted  to  those  who  have 
made  original  contributions  to  the  advancement  of  science  or 
learning,  Todhunter  was  one  of  those  whose  application  was 
granted  within  the  first  few  months."  In  1875  ne  published 
his  manual  Functions  of  Laplace,  Bessel  and  Legendre.  Next 
year  he  finished  an  arduous  literary  task — the  preparation 
of  two  volumes,  the  one  containing  an  account  of  the  writings 
of  Whewell,  the  other  containing  selections  from  his  literary 


146  TEN   BRITISH   MATHEMATICIANS 

and  scientific  correspondence.  Todhunter's  task  was  marred 
to  a  considerable  extent  by  an  unfortunate  division  of  the 
matter:  the  scientific  and  literary  details  were  given  to  him, 
while  the  writing  of  the  life  itself  was  given  to  another. 

In  the  summer  of  1880  Dr.  Todhunter  first  began  to  suffer 
from  his  eyesight,  and  from  that  date  he  gradually  and 
slowly  became  weaker.  But  it  was  not  till  September,  1883, 
when  he  was  at  Hunstanton;  that  the  worst  symptoms  came  on. 
He  then  partially  lost  by  paralysis  the  use  of  the  right  arm; 
and,  though  he  afterwards  recovered  from  this,  he  was  left 
much  weaker.  In  January  of  the  next  year  he  had  another 
attack,  and  he  died  on  March  i,  1884,  in  the  64th  year  of  his 
age. 

Todhunter  left  a  History  of  Elasticity  nearly  finished.  The 
manuscript  was  submitted  to  Cay  ley  for  report;  it  was  in  1886 
published  under  the  editorship  of  Karl  Pearson.  I  believe  that 
he  had  other  histories  in  contemplation;  I  had  the  honor  of 
meeting  him  once,  and  in  the  course  of  conversation  on  mathe- 
matical logic,  he  said  that  he  had  a  project  of  taking  up  the 
history  of  that  subject;  his  interest  in  it  dated  from  his  study 
under  De  Morgan.  Todhunter  had  the  same  ruling  passion  as 
Airy — love  of  order — and  was  thus  able  to  achieve  an  immense 
amount  of  mathematical  work.  Prof.  Mayor  wrote,  "  Tod- 
hunter had  no  enemies,  for  he  neither  coined  nor  circulated 
scandal;  men  of  all  sects  and  parties  were  at  home  with  him, 
for  he  was  many-sided  enough  to  see  good  in  every  thing.  His 
friendship  extended  even  to  the  lower  creatures.  The  canaries 
always  hung  in  his  room,  for  he  never  forgot  to  see  to  their 
wants." 


INDEX 


Adams,  J.  C.,  138 
Airy,  G.  B.,  38,  45,  146 
Apollonius,  102 
Argand,  J.  R.,  13,  56,  138 
Arnold,  T.,  93 

Babbage,  C.,  10,  13 

Ball,  R.,  14 

Beltrami,  E.,  86 

Boole,  G.,  50-63,  14,  29,  80,  98,  138 

Boscovich,  R.  J.,  130 

Brewster,  D.,  95 

Brinkley,  N.,  35,  36,  38 

Burkhardt,  J.  C.,  97 

Cauchy,  A.  L.,  95 

Cayley,  A.,  64-77,  46,  78, 108,  109,  no, 

121,  126,  132,  137,  143,  146 
Chrystal,  G.,  25,  35 
Clairault,  A.  C.,  10 
Clifford,  W.  K.,  78-91,  121 
Colburn,  Z.,  35 
Colenso,  J.  W.,  129 

Davy,  H.,  82,  114 

Delambre,  J.  B.  J.,  10 

DeLaRue,  W.,  118 

De  Morgan,  A.,  19-33,  8,  14,  4°,  4*, 
52,  S3,  54,  58,  62,  63,  65,  70,  78,  79, 
80,  108,  in,  123,  124,  132,  134,  135, 
138,  142,  146 

De  Morgan,  G.,  22 

Dewar,  J.,  82,  114 

Dodgson,  C.  L.,  80,  142 

Dodson,  J.,  19 


Eisenstein,  F.  G.,  99 
Ellis,  L.,  64 


Eratosthenes,  97 

Euclid,  19,  no,  in,  114,  140,  142 

Euler,  L.,  10,  96,  no 

Faraday,  M.,  82,  102,  114 
Fermat,  P.  de,  96 
Forsyth,  A.  R.,  53,  70 
Foucault,  J.  B.  L.,  141 
Frangois,  13 
Franklin,  F.,  116,  117 
Frend,  W.,  12,  13,  21 

Gauss,  K.  F.,  86,  95,  96,  100 

Graves,  C.,  54 

Green,  G.,  107 

Gregory,  D.,  52 

Gregory,  D.  F.,  14,  25,  27,  51,  52,  64, 

65,  i°7 
Gregory,  J.,  52 

Hamilton,  J.,  34 

Hamilton,  W.,  23,  29 

Hamilton,  W.  E.,  48 

Hamilton,  W.  R.,  34-49,  14,  18,  23,  28, 
52,  53,  54,  62,  69,  71,  73,  74,  75,  80, 
81,  85,  95,  114,  123,  127,  128,  132 

Harley,  R.,  54 

Hart,  H.,  115 

Helmholtz,  H.  L.  F.,  86 

Hermite,  C.,  105 

Herschel,  J.  F.  W.,  8,  10,  13,  88,  135 

High  ton,  H.,  93 

Hipparchus,  48 

Hobbes,  T.,  63 

Huxley,  T.  H.,  101,  in,  112,  113,  129 

Ivory,  J.,  10,  81 


147 


148 


INDEX 


Jacoby,  K.  G.  J.,  10,  99 
J°ly,  J.,  45,  49 

Kant,  E.,  41,  42,  74,  85,  87 
Kelvin,  Lord,  79 
Kempe,  A.  B.,  115,  116 
Kepler,  J.,  102 
Kirkman,  T.  P.,  122-133 

Lacroix,  S.  F.,  10 
Lagrange,  C.  F.  L.,  10,  40 
Laplace,  P.  S.,  10,  35,  65 
Legendre,  A.  M.,  10 
Leibnitz,  G.  W.,  9 
Little,  C.  N.,  133 
Lipkin,  114 
Lloyd,  H.,  41,  85 
Lobatchewsky,  N.  I.,  86 
Lockyer,  J.  N.,  101 
Lubbock,  J.,  101 

Macfarlane.    A.,   3,  4,  31,   45,  57,  67, 

101,  118,  133,  141,  146 
Maclaurin,  C.,  no 
Martineau,  J.,  22 
Maseres,  F.,  12,  14 

Maxwell,  J.  C.,  40,  67,  68,  79,  87,  101, 

102,  132,  140 
Mill,  J.  S.,  74,  129 
Minkowski,  105,  106 


Newton,  I.,  9,  41,  46,  65,  67,  80,  no 

Peacock,  G.,  7-18,  20,  24,  25,  41,  52, 

SS,78 
Peaucellier,  C.  N,,  114, 115 


Plato,  74,  91 
Poincarg,  J.  H.,  121 
Pollock,  F.,  81 

Rankine,  W.  J.  M.,  40 
Rayleigh,  Lord,  82 
Record,  R.,  16 
Riemann,  G.  F.  B.,  86 
Rosse,  Lord,  45 
Routh,  E.  J.,  136 
Rum  ford,  Count,  82 

Salmon,  G.,  46,  65,  143 
1  Smith,  H.  J.  S.,  92-106,  119,  120 

Smith,  J.,  32,  33 

Socrates,  24,  91 

Spencer,  H.,  129,  130,  131,  132 
;  Spottiswoode,  W.,  119 

Stewart,  B.,  89,  91 

Sylvester,  J.  J.,  107-121, 66, 69,  79, 135, 
HI,  143 

Tait,  P.  G.,  34,  42,  57,  71,  72,  73,  89,  91, 

132,  133,  136,  140,  143,  144,  146 
Todhunter,  I.,  134-146 
Tschebicheff,  114,  115 
Tyndall,  J.,  82,  114,  129 

Waring,  E.,  no 

Warren,  J.,  13,  28 

Watt,  J.,  115 

Whewell,  W.,  14,  20,  24,  29,  79,  94 

Woodhouse,  R.,  136 

Wordsworth,  W.,  38,  39 

Young,  T.,  82 


2486511 


THE  UNIVERSITY  LIBRARY 


UNIVERSITY  OF  CALIFORNIA,  SANTA  CRUZ 


SCIENCE  LIBRARY 


50m-4,'69(J7948s8)247' 


QC15.M3  Sci 


3  2106  00238  3435 


